Solving Quadratic Equations - Examples, Exercises and Solutions

To answer the question, we'll need to recall the quadratic formula:

$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$

Let's remember that:

a is the coefficient of X²

b is the coefficient of X

c is the free term

And if we look again at the formula given to us:

a=1

b=10

c=9

Let's substitute into the formula:

$x = {-10 \pm \sqrt{10^2-4\cdot 1 \cdot 9} \over 2\cdot 1}$

Let's start by solving what's under the square root:

$x = {-10 \pm \sqrt{100-36} \over 2}$

$x = {-10 \pm \sqrt{64} \over 2}$

$x = {-10 \pm 8 \over 2}$

Now we'll solve twice, once with plus and once with minus

$x = {-10 +8 \over 2}= {-2 \over 2} = -1$

$x = {-10 -8 \over 2} = {-18 \over 2} =-9$

And we can see that we got two solutions, X=-1 and X=-9

And that's the solution!

To solve the quadratic equation $-2X^2 + 6X + 8 = 0$ using the quadratic formula, follow these steps:

• Step 1: Calculate the discriminant, $b^2 - 4ac$.

Here, $a = -2$, $b = 6$, and $c = 8$. Plug these into the formula: $b^2 - 4ac = 6^2 - 4(-2)(8) = 36 + 64 = 100$ Since the discriminant is greater than zero, the roots are real and distinct.

• Step 2: Apply the quadratic formula, $X = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Substituting the values, we have: $X = \frac{-6 \pm \sqrt{100}}{2(-2)}$ Simplifying inside the square root gives us: $X = \frac{-6 \pm 10}{-4}$ This leads to two possible solutions: - First, calculate with the positive square root: $X_1 = \frac{-6 + 10}{-4} = \frac{4}{-4} = -1$ - Second, calculate with the negative square root: $X_2 = \frac{-6 - 10}{-4} = \frac{-16}{-4} = 4$

Thus, the solutions to the equation are $X_1 = 4$ and $X_2 = -1$.

Verifying against the choices, the correct choice is: : $X_1=4, X_2=-1$.

Therefore, the solution is $X_1 = 4, X_2 = -1$.

The parameters are expressed in the quadratic equation as follows:

aX2+bX+c=0

We substitute into the formula:

-5±√(5²-4*1*4) 
          2

-5±√(25-16)
         2

-5±√9
    2

-5±3
   2

The symbol ± means that we have to solve this part twice, once with a plus and a second time with a minus,

This is how we later get two results.

-5-3 = -8
-8/2 = -4

-5+3 = -2
-2/2 = -1

And thus we find out that X = -1, -4

To solve the quadratic equation $x^2 + 9x + 8 = 0$, we will use the factoring method because it appears simple to factor.

First, we attempt to factor the quadratic expression $x^2 + 9x + 8$. We look for two numbers that multiply to 8 (the constant term) and add up to 9 (the coefficient of the $x$ term).

These numbers are 1 and 8. So, we can write:

$x^2 + 9x + 8 = (x + 1)(x + 8) = 0$

Now, to find the solutions, we set each factor equal to zero:

1. $x + 1 = 0$ → $x = -1$
2. $x + 8 = 0$ → $x = -8$

Therefore, the solutions to the equation are $x_1 = -1$ and $x_2 = -8$.

Upon reviewing the multiple-choice answers, we find that the correct choice is the one that matches our solutions:

$x_1=-1$ $x_2=-8$

Let's recall the quadratic formula:

We'll substitute the given data into the formula:

$x={{-(-10)\pm\sqrt{-10^2-4\cdot2\cdot(-12)}\over 2\cdot2}}$

Let's simplify and solve the part under the square root:

$x={{10\pm\sqrt{100+96}\over 4}}$

$x={{10\pm\sqrt{196}\over 4}}$

$x={{10\pm14\over 4}}$

Now we'll solve using both methods, once with the addition sign and once with the subtraction sign:

$x={{10+14\over 4}} = {24\over4}=6$

$x={{10-14\over 4}} = {-4\over4}=-1$

We've arrived at the solution: X=6,-1

To solve the quadratic equation $3x^2 - 39x - 90 = 0$, we will use the quadratic formula.

• Step 1: Identify coefficients: $a = 3$, $b = -39$, and $c = -90$.
• Step 2: Calculate the discriminant: $b^2 - 4ac$.
• Step 3: Apply the quadratic formula to find $x_1$ and $x_2$.

Now, let's work through the steps:

Step 1: Coefficients are given as $a = 3$, $b = -39$, $c = -90$.

Step 2: The discriminant is calculated as follows:

$b^2 - 4ac = (-39)^2 - 4 \cdot 3 \cdot (-90) = 1521 + 1080 = 2601$.

The discriminant is positive, indicating two distinct real solutions.

Step 3: Apply the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-(-39) \pm \sqrt{2601}}{2 \cdot 3}$

This simplifies to:

$x = \frac{39 \pm 51}{6}$

Calculating the two solutions:

• $x_1 = \frac{39 + 51}{6} = \frac{90}{6} = 15$.
• $x_2 = \frac{39 - 51}{6} = \frac{-12}{6} = -2$.

Therefore, the solutions to the equation are $x_1 = 15$ and $x_2 = -2$.

Comparing with the choices, the correct answer is:

$x_1 = 15$ $x_2 = -2$

To solve this quadratic equation, we will use the quadratic formula. Let's go through the process step-by-step:

• Step 1: Identify the coefficients.

The coefficients are $a = -2$, $b = 22$, and $c = -60$.

• Step 2: Calculate the discriminant.

The discriminant $D$ is calculated using the formula $b^2 - 4ac$.
Here, $D = 22^2 - 4(-2)(-60) = 484 - 480 = 4$.

• Step 3: Apply the quadratic formula.

The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Substituting the values, we get $x = \frac{-22 \pm \sqrt{4}}{2(-2)}$.

• Step 4: Simplify and solve for $x$.

The expression inside the square root is $\sqrt{4} = 2$.
Therefore, we have two potential solutions:
$x_1 = \frac{-22 + 2}{-4} = \frac{-20}{-4} = 5$
$x_2 = \frac{-22 - 2}{-4} = \frac{-24}{-4} = 6$.

The solutions to the equation $-2x^2 + 22x - 60 = 0$ are $x_1 = 5$ and $x_2 = 6$.

In conclusion, the solution to the problem is:

$x_1=5$ $x_2=6$

To identify the coefficients from the quadratic equation $5x^2 + 6x - 8 = 0$, follow these steps:

• Standard Form of Quadratic Equation: Compare $5x^2 + 6x - 8$ with the standard form $ax^2 + bx + c = 0$.
• Identify $a$, $b$, and $c$:
  • $a = 5$: The coefficient of $x^2$.
  • $b = 6$: The coefficient of $x$.
  • $c = -8$: The constant term (independent of $x$).

Therefore, from the equation $5x^2 + 6x - 8 = 0$, the coefficients are identified as $a=5$, $b=6$, and $c=-8$.

Comparing with choices, we find that choice 2 is correct: $a=5$, $b=6$, $c=-8$.

Thus, the coefficients are identified as $a=5$, $b=6$, $c=-8$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the equation is given in the standard quadratic form: $-8x^2 - 5x + 9 = 0$.
• Step 2: Determine each component:
  • $a$ is the coefficient of $x^2$, which is $-8$.
  • $b$ is the coefficient of $x$, which is $-5$.
  • $c$ is the constant term, which is $9$.

Now, let's resolve this using the above plan:

Step 1: The equation is already in standard form: $-8x^2 - 5x + 9 = 0$.

Step 2: Recognize that:

• $a = -8$ because it is the coefficient of $x^2$.
• $b = -5$ because it is the coefficient of $x$.
• $c = 9$ because it is the constant term.

Therefore, the components of the equation are $a = -8$, $b = -5$, $c = 9$.

The quadratic equation we have is $x^2 + 4x - 5 = 0$.

We'll compare this with the general form of a quadratic equation: $ax^2 + bx + c = 0$.

1. Identify $a$: The coefficient of $x^2$ in the given equation is $1$. Therefore, $a = 1$.

2. Identify $b$: The coefficient of $x$ in the given equation is $4$. Therefore, $b = 4$.

3. Identify $c$: The constant term in the given equation is $-5$. Therefore, $c = -5$.

Thus, the components of the equation are:

• $a = 1$
• $b = 4$
• $c = -5$

The correct answer to this problem, matching choice id 3, is:

$a=1$ $b=4$ $c=-5$

To solve this problem, we need to identify the coefficients in the quadratic equation given by $-x^2 - 2 = 0$.

The standard form of a quadratic equation is:

$ax^2 + bx + c = 0$

In the given equation, $-x^2 - 2 = 0$, we can write it as:

$-1 \cdot x^2 + 0 \cdot x - 2 = 0$

This corresponds to the standard form with:

• Coefficient $a$: The coefficient of $x^2$ is $-1$.
• Coefficient $b$: The coefficient of $x$ is $0$, as there is no $x$ term.
• Coefficient $c$: The constant term is $-2$.

Upon examining the answer choices given, the correct choice must match these coefficients precisely.

The correct choice is:

$a = -1$, $b = 0$, $c = -2$

This matches Choice 3.

Let's solve the problem step-by-step:

First, consider the given equation:

$x^2 + 7x = 0$

This equation is almost in the standard form of a quadratic equation:

$ax^2 + bx + c = 0$

Where:

• $a$ is the coefficient of $x^2$
• $b$ is the coefficient of $x$
• $c$ is the constant term (independent number)

Let's identify each of these components from the given equation:

• For $a$: The term with $x^2$ is $x^2$. In this case, the coefficient is implicitly 1, so $a = 1$.
• For $b$: The term with $x$ is $7x$. The coefficient of $x$ is 7, so $b = 7$.
• For $c$: There is no independent constant term visible, so we assume $c = 0$.

Thus, the components of the quadratic equation are:

$a = 1$, $b = 7$, $c = 0$

The correct choice from the provided options is : $a=1$, $b=7$, $c=0$

To determine the components of the quadratic equation, follow these steps:

• Step 1: Recognize the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$.
• Step 2: Compare the given equation $10x^2 + 20x + 5 = 0$ to the standard form.
• Step 3: Identify the coefficients:
  - The term $10x^2$ indicates that $a = 10$.
  - The term $20x$ indicates that $b = 20$.
  - The term $5$ is the constant term, so $c = 5$.

Therefore, the components of the equation are:

$a = 10$, $b = 20$, $c = 5$.

The correct answer among the choices provided is the one that correctly identifies these coefficients:

$a=10$ $b=20$ $c=5$

Therefore, the correct choice is Choice 4.

Let's solve this problem step-by-step by identifying the coefficients of the quadratic equation:

First, examine the given equation:

$5 - 6x^2 + 12x = 0$

To make it easier to identify the coefficients, we rewrite the equation in the standard quadratic form:

$-6x^2 + 12x + 5 = 0$

In this expression, we can now directly identify the coefficients:

• The coefficient of $x^2$ (quadratic term) is $a = -6$.
• The coefficient of $x$ (linear term) is $b = 12$.
• The constant term (independent number) is $c = 5$.

Thus, the components of the quadratic equation are:

$a = -6$, $b = 12$, $c = 5$

By comparing these values to the multiple-choice options, we can determine that the correct choice is:

Choice 4: $a = -6$, $b = 12$, $c = 5$

Therefore, the final solution is:

$a = -6$, $b = 12$, $c = 5$.

Let's recall the general form of the quadratic function:

$y=ax^2+bx+c$ The function given in the problem is:

$y=-x^2+25x$$c$is the free term (meaning the coefficient of the term with power 0),

In the function in the problem there is no free term,

Therefore, we can identify that:

$c=0$Therefore, the correct answer is answer A.