The Quadratic Formula: System of equations with no solution

The parameters are expressed in the quadratic equation as follows:

aX2+bX+c=0

We identify that we have:
a=1
b=0
c=9

We recall the root formula:

We replace according to the formula:

-0 ± √(0²-4*1*9)

           2

We will focus on the part inside the square root (also called delta)

√(0-4*1*9)

√(0-36)

√-36

It is not possible to take the square root of a negative number.

And so the question has no solution.

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

• Step 1: Calculate the discriminant of the quadratic equation.
• Step 2: Use the discriminant to determine the number and type of solutions.

Step 1: Calculate the discriminant using the formula:

$\Delta = b^2 - 4ac = 5^2 - 4 \times 1 \times 10 = 25 - 40 = -15$.

Step 2: Analyze the discriminant:

• Since the discriminant ($\Delta$) is negative ($-15$), this indicates that the quadratic equation has no real solutions.

Therefore, the final solution is that the equation $x^2 + 5x + 10 = 0$ has no solution.

Comparing this with the given answer choices, the correct choice is:

Choice 3: No solution

Let's solve the quadratic equation $-5x^2 - 2x - 8 = 0$ using the quadratic formula:

First, identify the coefficients: $a = -5$, $b = -2$, and $c = -8$.

The quadratic formula is given by:

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

We start by calculating the discriminant, $D = b^2 - 4ac$:

$D = (-2)^2 - 4 \cdot (-5) \cdot (-8)$

$D = 4 - 160$

$D = -156$

Since the discriminant $D = -156$ is less than zero, this indicates that there are no real solutions for the quadratic equation $-5x^2 - 2x - 8 = 0$.

Therefore, the equation has no solution in terms of real numbers.

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

• Step 1: Identify the coefficients $a = 3$, $b = -9$, $c = 12$.
• Step 2: Compute the discriminant $\Delta = b^2 - 4ac$.
• Step 3: Determine if real solutions exist based on the discriminant.
• Step 4: Solve using the Quadratic Formula if applicable.

Now, let's work through each step:

Step 2: Calculate the discriminant:

$\Delta = (-9)^2 - 4 \times 3 \times 12$

$\Delta = 81 - 144 = -63$

Step 3: Since the discriminant $\Delta$ is negative ($-63$), this means there are no real solutions for the quadratic equation. A negative discriminant indicates that the solutions are complex (i.e., non-real), so the equation has no real solution.

Therefore, the solution to the problem is No solution.

To solve the quadratic equation $-6x^2 + 12x - 14 = 0$, we'll use the quadratic formula:

The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Let's calculate step by step:

• Identify the coefficients: Here, $a = -6$, $b = 12$, and $c = -14$.
• Compute the discriminant: $b^2 - 4ac = 12^2 - 4(-6)(-14) = 144 - 336 = -192$.
• Assess the discriminant: The discriminant is $-192$, which is less than zero.
• Since the discriminant is negative, there are no real solutions to the equation.

Therefore, the correct answer to the problem is "No solution."

To solve the quadratic equation $x^2 + 3x + 7 = 0$, we will use the quadratic formula:

• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• Identify the coefficients: $a = 1$, $b = 3$, $c = 7$.
• Calculate the discriminant: $D = b^2 - 4ac = 3^2 - 4(1)(7) = 9 - 28 = -19$.

The discriminant $D = -19$ is negative. This indicates that there are no real roots to the equation. Quadratic equations with negative discriminants have complex solutions.

Therefore, the given quadratic equation $x^2 + 3x + 7 = 0$ has no solutions in the real number system, and we conclude that:

No solution.

To solve the quadratic equation $x^2 - 2x + 8 = 0$, we'll apply the quadratic formula method. The process is as follows:

• Step 1: Identify the coefficients from the equation: $a = 1$, $b = -2$, $c = 8$.
• Step 2: Calculate the discriminant using the formula $\Delta = b^2 - 4ac$.
• Step 3: For the given equation, calculate $\Delta = (-2)^2 - 4(1)(8) = 4 - 32 = -28$.
• Step 4: Interpret the discriminant. Since $\Delta = -28$ is negative, this indicates there are no real solutions to the equation.

Without real roots, the equation has complex solutions. Since the problem requires real solutions, we conclude:

The equation $x^2 - 2x + 8 = 0$ has no real solution.

The following is a quadratic equation:

$2x^2-3x+5=0$

This is due to the fact that there is a quadratic term (meaning raised to the second power),

The first step in solving a quadratic equation is always arranging it in a form where all terms on one side are ordered from highest to lowest power (in descending order from left to right) and 0 on the other side,

Then we can choose whether to solve it using the quadratic formula or by factoring/completing the square.

The equation in the problem is already arranged, so let's proceed to solve it using the quadratic formula,

Remember:

The rule states that the roots of an equation in the form:

$ax^2+bx+c=0$

are:

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

(meaning its solutions, the two possible values of the unknown for which we get a true statement when substituted in the equation)

$$This formula is called: "The Quadratic Formula"

Let's return to the problem:

$2x^2-3x+5=0$and solve it:

First, let's identify the coefficients of the terms:

$\begin{cases}a=2 \\ b=-3 \\ c=5\end{cases}$

where we noted that the coefficient includes the minus sign, and this is because in the general form of the equation we mentioned earlier:

$ax^2+bx+c=0$

the coefficients are defined such that they have a plus sign in front of them, and therefore the minus sign must be included in the coefficient value.

Let's continue and obtain the equation's solutions (roots) by substituting the coefficients we noted earlier in the quadratic formula:

$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{-(-3)\pm\sqrt{(-3)^2-4\cdot2\cdot5}}{2\cdot2}$

Let's continue and calculate the expression under the root and simplify the expression:

$x_{1,2}=\frac{3\pm\sqrt{-31}}{2}\frac{}{}$

The expression under the root is negative, and since we cannot extract a real root from a negative number, this equation has no real solutions,

Meaning - there is no real value of $x$that when substituted in the equation will give a true statement.

Therefore, the correct answer is answer D.

This is a quadratic equation:

$-x^2+6x-10=0$

due to the fact that there is a quadratic term present(meaning raised to the second power),

The first step in solving a quadratic equation is always arranging it in a form where all terms on one side are ordered from highest to lowest power (in descending order from left to right) and 0 on the other side,

Then we can choose whether to solve it using the quadratic formula or by factoring/completing the square.

The equation in the problem is already arranged, so let's proceed with the solving technique:

For ease of solving and minimizing errors, it is always recommended to ensure that the coefficient of the quadratic term in the equation is positive,

We'll achieve this by multiplying (both sides of) the equation by:$-1$:

$-x^2+6x-10=0 \hspace{8pt}\text{/}\cdot(-1)\\ x^2-6x+10=0$$$

Let's continue solving the equation

Solve it using the quadratic formula,

Remember:

The rule states that the roots of an equation of the form:

$ax^2+bx+c=0$

are:

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

(meaning its solutions, the two possible values of the unknown for which we get a true statement when substituted in the equation)

$$This formula is called: "The Quadratic Formula"

Let's return to the problem:

$x^2-6x+10=0$

and solve it:

First, let's identify the coefficients of the terms:

$\begin{cases}a=1 \\ b=-6 \\ c=10\end{cases}$

where we noted that the coefficient of the quadratic term is 1,

We obtain the equation's solutions (roots) by substituting these coefficients that we mentioned earlier in the quadratic formula:

$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{-(-6)\pm\sqrt{(-6)^2-4\cdot1\cdot10}}{2\cdot1}$

Let's continue and calculate the expression under the root and simplify the expression:

$x_{1,2}=\frac{6\pm\sqrt{-4}}{2}$

The expression under the root is negative, and since we cannot extract a real root from a negative number, this equation has no real solutions,

Meaning - there is no real value of $x$that when substituted in the equation will give a true statement.

Therefore, the correct answer is answer D.

To solve the quadratic equation $-2x^2 - 5x - 9 = 0$, we follow these steps:

• Step 1: Identify coefficients $a = -2$, $b = -5$, $c = -9$.
• Step 2: Compute the discriminant, $b^2 - 4ac$.
• Step 3: Analyze the discriminant's value to determine the nature of the roots.

Step 1: We have $a = -2$, $b = -5$, $c = -9$.

Step 2: Calculate the discriminant:
$b^2 - 4ac = (-5)^2 - 4(-2)(-9) = 25 - 72 = -47$.

Step 3: Since the discriminant value is $-47$, which is less than zero, this indicates that there are no real solutions because the square root of a negative number is imaginary.

Therefore, since there are no real solutions, we conclude that the problem has no real solutions.

Thus, the solution to the equation is No solution.

Let's solve the quadratic equation $5x^2 + 1.5x + 1 = 0$ using the quadratic formula.

Step 1: Identify the coefficients for the equation $ax^2 + bx + c = 0$:
$a = 5$
$b = 1.5$
$c = 1$

Step 2: Calculate the discriminant $\Delta = b^2 - 4ac$.
$\Delta = (1.5)^2 - 4 \cdot 5 \cdot 1 = 2.25 - 20 = -17.75$

Step 3: Analyze the discriminant:
Since the discriminant $\Delta = -17.75$ is negative, this indicates that there are no real solutions to the equation $5x^2 + 1.5x + 1 = 0$.

Therefore, the equation has no solution in the real number system.

To solve this problem, we'll apply the quadratic formula. The steps are as follows:

• Identify the coefficients: $a = 1$, $b = 3$, $c = 2.5$.
• Calculate the discriminant, $\Delta = b^2 - 4ac$.
• Apply the quadratic formula if the discriminant is non-negative.

Let's evaluate the discriminant:
Discriminant, $\Delta = b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot 2.5 = 9 - 10 = -1$.

The discriminant is $-1$, which is less than zero. This means the quadratic equation has no real solutions. Complex solutions are not considered here based on the problem context.

Therefore, the solution to the problem is No solution.

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

• Step 1: Identify the provided equation and standardize it.
• Step 2: Determine the coefficients $a$, $b$, and $c$.
• Step 3: Compute the discriminant $b^2 - 4ac$.
• Step 4: Analyze the discriminant to determine the nature of the solutions.

Let's work through each step:

Step 1: The given equation is $\frac{x^2}{2} - x + \frac{2}{3} = 0$. For simplicity, we multiply through by 6 to clear fractions:
This becomes $3x^2 - 6x + 4 = 0$.

Step 2: Identify coefficients for the quadratic formula:
$a = 3$, $b = -6$, and $c = 4$.

Step 3: Compute the discriminant $b^2 - 4ac$:
Discriminant $= (-6)^2 - 4 \times 3 \times 4 = 36 - 48 = -12$.

Step 4: Analyze the discriminant:
The discriminant is negative ($-12$), indicating no real solutions.

No solution exists for this equation in the real number set.

Therefore, the solution to the problem is: No solution.

To solve the quadratic equation $\frac{x^2}{2} + \frac{x}{4} + 1 = 0$, follow these steps:

• Step 1: Convert to Standard Form
  Begin by eliminating the fractions to simplify. Multiply the entire equation by 4, the least common multiple of the denominators: $4 \left( \frac{x^2}{2} + \frac{x}{4} + 1 \right) = 4 \cdot 0$ which simplifies to: $2x^2 + x + 4 = 0$ Now, it's in standard quadratic form: $ax^2 + bx + c = 0$, with $a = 2$, $b = 1$, and $c = 4$.
• Step 2: Evaluate the Discriminant
  The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is calculated as: $b^2 - 4ac$ Substituting the values, we have: $1^2 - 4 \cdot 2 \cdot 4 = 1 - 32 = -31$ Since the discriminant is negative, it indicates that there are no real solutions for the equation.

Conclusion: The given quadratic equation has no real solutions due to the negative discriminant.

The correct answer to the problem is therefore, No solution.

To solve the equation $7x^2 + 3x + 8 = 9x + 3$, follow these steps:

• Step 1: Rearrange the equation into standard quadratic form:
  Move all terms to one side:
  $7x^2 + 3x + 8 - 9x - 3 = 0$.
• Step 2: Simplify the equation:
  Combine like terms:
  $7x^2 - 6x + 5 = 0$.
• Step 3: Identify coefficients:
  $a = 7$, $b = -6$, and $c = 5$.
• Step 4: Calculate the discriminant ($\Delta$):
  $\Delta = b^2 - 4ac = (-6)^2 - 4(7)(5) = 36 - 140 = -104$.
• Step 5: Determine the nature of the roots:
  Since the discriminant is negative ($\Delta = -104$), this means there are no real solutions.

Therefore, the solution to the equation is No solution.

Given the equation:

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

Step 1: Move all terms to one side:

Subtract $2x^2$ from both sides to get:

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

Simplify this to:

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

Step 2: Rearrange to standard form:

Multiply the entire equation by -1 for simplicity:

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

Step 3: Solve the quadratic equation using the quadratic formula:

Here, $a = 1$, $b = -3$, $c = 4$.

Plug into the quadratic formula:

$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \times 1 \times 4}}{2 \times 1}$

$x = \frac{3 \pm \sqrt{9 - 16}}{2}$

$x = \frac{3 \pm \sqrt{-7}}{2}$

Step 4: Interpret the result:

The discriminant ($b^2 - 4ac$) is negative, $-7$, indicating no real solutions.

Conclusion: The equation has no solution in the set of real numbers.

In comparison with the provided choices, the correct choice is:

No solution