Factoring Trinomials: How many solutions exist to a problem?

Let's solve the given equation:

$x^2+4x+3=0$

We identify that the coefficient of the quadratic term is 1, therefore we can (try to) solve it using the quick trinomial method,

We'll look for a pair of numbers whose product is the free term in the left side expression, and whose sum is the coefficient of the first power term meaning two numbers $m,\hspace{2pt}n$ that satisfy:

$m\cdot n=3\\ m+n=4$

From the first requirement above, meaning - from the multiplication, we can conclude according to the rules of sign multiplication that both numbers have the same sign, and now we'll remember that the possible factor pairs of the (prime) number 3 are 3 and 1, fulfilling the second requirement mentioned, together with the fact that the signs of the numbers we're looking for are equal to each other will lead to the conclusion that the only possibility for the two numbers we're looking for is:

$\begin{cases} m=3\\ n=1 \end{cases}$

and therefore we'll factor the expression on the left side of the equation to:

$x^2+4x+3=0\\ \downarrow\\ (x+3)(x+1)=0$

where we used the pair of numbers we found earlier in this factorization,

We'll continue and consider the fact that in the left side of the equation we got in the last step there is a multiplication of algebraic expressions and on the right side the number 0, therefore, since the only way to get 0 from multiplication is to multiply by 0, at least one of the expressions in the multiplication on the left side must equal zero,

meaning:

$x+3=0\\ \boxed{x=-3}$

or:

$x+1=0\\ \boxed{x=-1}$

Therefore the given equation has two real solutions.

Therefore the correct answer is answer B.

Important note:

Note that we could answer the question asked in the problem and find the number of real solutions to the equation (and if there are any real solutions to this equation at all), without solving it,

For this we'll recall the roots formula:

The rule states that for a quadratic equation in the general form:

$ax^2+bx+c =0$there are two solutions (or fewer) which we find using the formula:

$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

We can now denote the expression under the root in the general equation solutions as:

$\Delta$ (The Greek letter delta), meaning, mathematically:

$x_{1,2}=\frac{-b\pm\sqrt{\textcolor{orange}{\Delta}}}{2a} \longleftrightarrow\textcolor{orange}{\Delta}=b^2-4ac$

We can now recall that a real root can only be calculated for a positive number or zero (since you cannot get a negative number by multiplying a real number by itself), and therefore an equation has two real solutions (or one real solution) only if:

$\Delta\geq0$

Later we'll note that if: $\Delta=0$ then the only solution to the equation is:

$x=-\frac{b}{2a}$

Therefore, we can summarize what was explained into a rule:

In a quadratic equation:

$ax^2+bx+c =0$

where we extract the coefficients and calculate delta:

$\Delta=b^2-4ac$

If:

a. \Delta<0 :

There is no (real) solution to the equation.

b.$\Delta=0$:

There is one (real) solution to the equation.

c.$\Delta=0$:

There are two (real) solutions to the equation.

Now let's return to the given equation and extract from it the coefficients:

$x^2+4x+3=0 \longrightarrow\begin{cases}a=1\\b=4\\c=3\end{cases}$We'll continue and calculate $\Delta$:

$\Delta=b^2-4ac\longrightarrow \Delta=4^2-4\cdot1\cdot3=4\\ \downarrow\\ \boxed{\Delta=4}$meaning we got that:

\boxed{ \Delta>0} Therefore the given equation has two real solutions.

Let's solve the given equation:

$x^4+12x^3+36x^2=0$We note that it is possible to factor the expression which is in the left side of the given equation, this is done by taking out the common factor $x^2$ which is the greatest common factor of the numbers and letters in the expression:

$x^4+12x^3+36x^2=0 \\ \downarrow\\ x^2(x^2+12x+36) =0$We will focus on the left side of the equation and then on the right side (the number 0).

Since the only way to get the result 0 from a product is to multiply by 0, at least one of the expressions in the product on the left side, must be equal to zero,

Meaning:

$x^2=0\hspace{6pt}\text{/}\sqrt{\hspace{4pt}}\\ \boxed{x=0}$

Or:

$x^2+12x+36=0$In order to find the additional solutions to the equation we must solve the equation:

Note that the first coefficient is 1, so we can try to solve it using the trinomial formula.

However, we can factor, in this case, also using the short multiplication formula for a binomial:

$(\textcolor{red}{a}+\textcolor{blue}{b})^2 = \textcolor{red}{a}^2+2\textcolor{red}{a}\textcolor{blue}{b}+\textcolor{blue}{b}^2$The reason for trying factoring in this approach is that we can identify in the left side of the equation we got in the last step, that the two terms which are in the far sides (meaning the term in the first position - it is the squared term and the term in the zero position - it is the free number in the expression) can be presented (simply) as a squared term:

$x^2+12x+36=0 \\ \downarrow\\ x^2+12x+6^2=0$

Equating the expression on the left side in the equation:

$\downarrow\\ x^2+12x+6^2$

To the expression on the right side in the short formula above:

$\textcolor{red}{a}^2+2\textcolor{red}{a}\textcolor{blue}{b}+\textcolor{blue}{b}^2$

The conclusion from this is that what remains to check is whether the middle term in the equation matches the middle term in the short multiplication formula above, meaning - after identifying $a$$b$which are both in the first position in the short multiplication formula above in which $a$and $b$we check if the middle term in the expression in the left side of the equation can be presented as $2\cdot a \cdot b$So, we start by presenting the equation of the short formula to the given expression:

$\textcolor{red}{a}^2+\underline{2\textcolor{red}{a}\textcolor{blue}{b}}+\textcolor{blue}{b}^2 \Leftrightarrow \textcolor{red}{x}^2+\underline{12x}+\textcolor{blue}{6}^2$And indeed it holds that:

$2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{6}=12x$Meaning the middle term in the expression in the equation indeed matches the form of the middle term in the short multiplication formula (highlighted with a line below), mathematically:

$x^2+\underline{12x}+6^2=0 \\ \textcolor{red}{x}^2+\underline{2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{6}}+\textcolor{blue}{6}^2=0\\ \downarrow\\ (\textcolor{red}{x}+\textcolor{blue}{6})^2=0$We can now remember that a real root can be calculated only for a positive number or for the number zero (since it is not possible to get a negative number from squaring a real number itself), and therefore for an equation there are two real solutions (or one solution) only if:

$$Next we note that if: $(x+6)^2=0\hspace{6pt}\text{/}\sqrt{\hspace{4pt}}\\ x+6=0\\ \boxed{x=-6}$then the only solution to the equation is:

$x^4+12x^3+36x^2=0 \\ \downarrow\\ x^2(x^2+12x+36) =0 \\ \downarrow\\ x^2=0\rightarrow\boxed{x=0} \\ x^2+12x+36=0\\ x^2+2\cdot2\cdot6+6^2=0\\ \rightarrow(x+6)^2=0\rightarrow\boxed{x=-6}\\ \downarrow\\ \boxed{x=0,-6}$Therefore, we can summarize what was explained using the following:

In the quadratic equation:

$ax^2+bx+c =0$in which the coefficients are substituted and the discriminant is calculated:

$a,b$If it holds:

a.$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$:

There is no (real) solution to the equation.

b.$\Delta$:

There exists a single (real) solution to the equation.

c.$x_{1,2}=\frac{-b\pm\sqrt{\textcolor{orange}{\Delta}}}{2a} \longleftrightarrow\textcolor{orange}{\Delta}=b^2-4ac$:

There exist two (real) solutions to the equation.

Now let's return to the given equation and extract from it the coefficients:

$\Delta\geq0$We continue and calculate $\Delta=0$:

$x=-\frac{b}{2a}$Therefore for the quadratic equation that we solved, one (real) solution,

and in combination with the solution $ax^2+bx+c =0$(the additional solution we found for the given equation which is indicated in the first step after factoring using the common factor),

Therefore we get that for the given equation:

$\Delta=b^2-4ac$

two real solutions.

In the given equation:

$x^3+1=0$The simplest and fastest way to find the number of its solutions,

will be simply to solve it, we will do this by moving terms to isolate the unknown, then we will take the cube root of both sides of the equation, while remembering that an odd root preserves the sign of the expression inside the root (meaning - the minus sign can be taken out of an odd root):

$x^3+1=0 \\ x^3=-1\hspace{6pt}\text{/}\sqrt[3]{\hspace{4pt}}\\ \downarrow\\ \sqrt[3]{x^3}=\sqrt[3]{-1}\\ x=-\sqrt[3]{1}\\ \boxed{x=-1}$meaning the given equation has a single solution,

therefore the correct answer is answer A.

In the given equation:

$x^3-x^2-4x+4=0$

The simplest and fastest way to find the number of solutions,

will be simply to solve it, noting that the expression on the left side contains four different terms,

Additionally, note that we cannot factor out a common factor for all terms in the expression, since there is a free term,

therefore we will turn to factoring by groups:

We will consider two groups of terms in the given expression so that each group has two terms, we will choose the groups so that in each group only one term has an extreme power (in this case the third power and zero power - of the free term, are the extreme powers in the expression and therefore we will include in each group only one of these terms):

$x^3-x^2-4x+4=0 \\ \downarrow\\ \textcolor{red}{x^3-x^2}\textcolor{blue}{-4x+4}=0$

We'll continue and note that each group marked separately, can be factored by taking out a common factor, first we'll factor out a common factor from the first group, marked in red, and continue to the second group, marked in blue, we'll also factor it by taking out a common factor so that the expression inside the parentheses will be identical to the expression in parentheses in the second group (marked in red) in the following calculation the identical expression in parentheses will be emphasized with an underline:

$\textcolor{red}{x^3-x^2}\textcolor{blue}{-4x+4}=0 \\ \downarrow\\ \textcolor{red}{x^2\underline{(x-1)}}\textcolor{blue}{-4x+4}=0 \\ \downarrow\\ \textcolor{red}{x^2\underline{(x-1)}}\textcolor{blue}{-4\underline{(x-1)}}=0 \\$We'll continue and note that now- the complete expression on the left side can be factored further by taking out a common factor that is a binomial, meaning- we'll factor out the identical expression in parentheses that was marked with an underline, as a common factor, we'll do this:

$x^2\underline{(x-1)}-4\underline{(x-1)}=0 \\ \downarrow\\ \underline{(x-1)}(x^2-4)=0$We'll continue and note that the right expression in the product of expressions obtained on the left side, can also be factored,

this- using the perfect square binomial formula:

$a^2-b^2=(a+b)(a-b)$

Let's return to the equation we got and do this:

$(x-1)(x^2-4)=0 \\ (x-1)\textcolor{orange}{(x^2-2^2)}=0 \\ \downarrow\\ (x-1)\textcolor{orange}{(x+2)(x-2)}=0 \\$We got on the left side a product of expressions that must equal zero, remember that a product of expressions equals zero if and only if at least one of the expressions equals 0, therefore we get three simpler equations and solve them:

$x-1=0\\ \downarrow\\ \boxed{x=1}$

or:

$x+2=0\\ \downarrow\\ \boxed{x=-2}$

or:

$x-2=0\\ \downarrow\\ \boxed{x=2}$

Let's summarize then the equation solving steps:

$x^3-x^2-4x+4=0 \\ \downarrow\\ \textcolor{red}{x^3-x^2}\textcolor{blue}{-4x+4}=0 \\ \downarrow\\ \textcolor{red}{x^2\underline{(x-1)}}\textcolor{blue}{-4\underline{(x-1)}}=0 \\ \downarrow\\ \underline{(x-1)}\textcolor{orange}{(x^2-4)}=0 \\ (x-1)\textcolor{orange}{(x+2)(x-2)}=0 \\ \downarrow\\ x-1=0\rightarrow\boxed{x=1}\\ x+2=0\rightarrow\boxed{x=-2}\\ x-2=0\rightarrow\boxed{x=2}\\ \downarrow\\ \boxed{x=1,-2,2}$

Meaning- the given equation has three solutions,

Therefore the correct answer is answer C.

Let's solve the given equation:

$x^4-5x^2+4=0$

We identify that this is a bi-quadratic equation that can be easily solved using substitution of a new variable,

That is, let's notice that:

$(x^2)^2=x^4$

Therefore, we can write the given equation in the following form:$\textcolor{blue}{ x^4}-5x^2+4=0 \\ \downarrow\\ \textcolor{blue}{ (x^2)^2}-5x^2+4=0$

Now let's define a new variable, $t$, such that:

$t=x^2$

If we substitute this new variable, $t$, in the given equation instead of $x^2$ we'll obtain an equation that depends only on $t$:

$(\textcolor{red}{x^2})^2-5\textcolor{red}{x^2}+4=0 \\ \downarrow\downarrow \boxed{\textcolor{red}{(x^2=t)}}\\ t^2-5t+4=0$

Proceed to solve the new equation that we obtained for the variable $t$. After we determine the values of variable t for which the equation holds, we'll go back and substitute each of them into the definition of t that we mentioned before in order to determine the value of x,

We identify that the equation that we obtained in the last step for t is a quadratic equation that can be solved using quick trinomial factoring:

$t^2-5t+4=0 \longleftrightarrow\begin{cases}\underline{?}\cdot\underline{?}=4\\ \underline{?}+\underline{?}=-5\end{cases}\\ \downarrow\\ (t-4)(t-1)=0$

Therefore we'll obtain two simpler equations from which we'll extract the solution for t:

$(t-4)(t-1)=0 \\ \downarrow\\ t-4=0\rightarrow\boxed{t=4}\\ t-1=0\rightarrow\boxed{t=1}\\ \boxed{t=1,4}$

Now let's go back to the definition of t that was mentioned before, let's recall it:

$t=x^2$ Notice that given the power of x is even, the variable t can get only non-negative values (meaning positive or zero),

Therefore the two values that we obtained for t from solving the quadratic equation are indeed valid,

We'll continue to substitute each of the two values we that we obtained for t in the definition of t mentioned before to solve the equation and then proceed to extract the corresponding value of x by solving the resulting equation using square root on both sides:

$\boxed{x^2=t}\\ \downarrow\\ t=1\textcolor{blue}{\rightarrow} x^2=1\hspace{6pt}\text{/}\sqrt{\hspace{4pt}}\textcolor{blue}{\rightarrow} \boxed{x=\pm1}\\ t=4\textcolor{blue}{\rightarrow} x^2=4\hspace{6pt}\text{/}\sqrt{\hspace{4pt}}\textcolor{blue}{\rightarrow} \boxed{x=\pm2}\\ \downarrow\\ \boxed{x=\pm1,\pm2}$

Let's summarize the steps of solving the equation:

$x^4-5x^2+4=0 \\ \downarrow\\ (\textcolor{red}{x^2})^2-5\textcolor{red}{x^2}+4=0 \\ \downarrow\downarrow \boxed{\textcolor{red}{(x^2=t)}}\\ t^2-5t+4=0 \\ \downarrow\\ (t-4)(t-1)=0 \\ \downarrow\\ t-4=0\rightarrow\boxed{t=4}\\ t-1=0\rightarrow\boxed{t=1}\\ \boxed{t=1,4}\\ \updownarrow\updownarrow\\ \boxed{x^2=t}\\ \downarrow\\ t=1\textcolor{blue}{\rightarrow} x^2=1\hspace{6pt}\text{/}\sqrt{\hspace{4pt}}\textcolor{blue}{\rightarrow} \boxed{x=\pm1}\\ t=4\textcolor{blue}{\rightarrow} x^2=4\hspace{6pt}\text{/}\sqrt{\hspace{4pt}}\textcolor{blue}{\rightarrow} \boxed{x=\pm2}\\ \downarrow\\ \boxed{x=\pm1,\pm2}$

Therefore the given equation has 4 different solutions,

Which means the correct answer is answer D.

Let's observe that the given equation:

$x^2+10x+9=0$is a quadratic equation that can be solved using quick factoring:

$x^2+10x+9=0 \longleftrightarrow\begin{cases}\underline{?}\cdot\underline{?}=9\\ \underline{?}+\underline{?}=10\end{cases}\\ \downarrow\\ (x+1)(x+9)=0$and therefore we get two simpler equations from which we can extract the solution:

$(x+1)(x+9)=0\\ \downarrow\\ x+1=0\rightarrow\boxed{x=-1}\\ x+9=0\rightarrow\boxed{x=-9}\\ \boxed{x=-1,-9}$and therefore the given equation has two solutions,

Thus, the correct answer is answer B.

Let's solve the given equation:

$x^3-2x^2+x=0$

Note that we can factor the expression on the left side by factoring out the common factor:

$$$x^3-2x^2+x=0 \\ \downarrow\\ x(x^2-2x+1)=0$

Proceed to factor the expression inside of the parentheses. It can be factored by using the perfect square trinomial formula:

$(a\pm b)^2=a^2\pm2ab+b^2$

As shown below:

$x(x^2-2x+1)=0 \\ x(x^2 \textcolor{blue}{-2x}+1^2)=0 \\ x(x^2\textcolor{blue}{-2\cdot x \cdot 1}+1^2)=0 \\ \downarrow\\ x(x-1)^2=0$

We should emphasize that the process of factoring by using the mentioned formula was only possible due to the middle term in the expression. (The first power in this case is highlighted in blue indeed matched the middle term in the perfect square trinomial formula)

Having obtained two simpler equations let's proceed to solve them:

$x(x-1)^2=0 \\ \boxed{x=0}\\ (x-1)^2=0\hspace{6pt}\text{/}\sqrt{\hspace{4pt}}\rightarrow x-1=0\rightarrow \boxed{x=1}\\ \downarrow\\ \boxed{x=0,1}$

Shown below is a summary of the various steps to solve the given equation:

$$$x^3-2x^2+x=0 \\ \downarrow\\ x(x^2-2x+1)=0 \\ \downarrow\\ x(x-1)^2=0 \\ \downarrow\\ \boxed{x=0}\\ (x-1)^2=0\rightarrow \boxed{x=1}\\ \downarrow\\ \boxed{x=0,1}$

Therefore, the given equation has two different solutions,

Which means - the correct answer is answer B.