Problem 68 Use the quadratic formula and a ... [FREE SOLUTION]

Chapter 8: Problem 68

Use the quadratic formula and a calculator to solve each equation. Round answers to three decimal places and check your answers. $$x^{2}+7.15 x+3.24=0$$

Short Answer

Expert verified

The solutions are $x \approx -0.487$ and $x \approx -6.664$.

Step by step solution

Identify coefficients

For the quadratic equation in the standard form $ax^2 + bx + c = 0$, identify the values of the coefficients. In this equation, $a = 1$, $b = 7.15$, and $c = 3.24$.

Write the quadratic formula

The quadratic formula to solve for $x$ is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Calculate the discriminant

Calculate the discriminant inside the square root: \[\Delta = b^2 - 4ac\]. Substitute $a = 1$, $b = 7.15$, and $c = 3.24$: \[\Delta = (7.15)^2 - 4 \cdot 1 \cdot 3.24 = 51.1225 - 12.96 = 38.1625\]

Calculate the square root of the discriminant

Find the square root of the discriminant: \[ \sqrt{38.1625} \approx 6.177 \]

Apply the quadratic formula

Now use the quadratic formula with the positive and negative square root results: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \] Substitute $b = 7.15$, $\sqrt{\Delta} = 6.177$, and $a = 1$: \[x = \frac{-7.15 \pm 6.177}{2}\]

Solve for the two values of x

Calculate the two solutions from the formula: \[ x_1 = \frac{-7.15 + 6.177}{2} = \frac{-0.973}{2} \approx -0.487 \] \[ x_2 = \frac{-7.15 - 6.177}{2} = \frac{-13.327}{2} \approx -6.664 \]

Verify the solutions

Substitute the solutions back into the original equation to verify: For $ x_1 = -0.487 $: \[ (-0.487)^2 + 7.15(-0.487) + 3.24 \approx 0 \] For $ x_2 = -6.664 $: \[ (-6.664)^2 + 7.15(-6.664) + 3.24 \approx 0 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discriminant

The discriminant is a key concept in solving quadratic equations using the quadratic formula. It is the part of the formula under the square root: $ \Delta = b^2 - 4ac $.
The discriminant ( $\Delta $ ) tells us valuable information about the roots of the quadratic equation:

If $\Delta $ > 0, there are two real and distinct solutions.
If $\Delta $ = 0, there is exactly one real solution (or a repeated root).
If $\Delta $ < 0, there are no real solutions, but two complex solutions.

In our exercise, the discriminant is calculated as follows: $\Delta = (7.15)^2 - 4 \cdot 1 \cdot 3.24 = 38.1625$.Since $\Delta$ is greater than zero, we know there are two real solutions. Discriminant helps us determine the nature of solutions even before we solve the equation fully.

Solving Quadratic Equations

To solve quadratic equations, we use the quadratic formula. The standard quadratic equation is: $ax^2 + bx + c = 0$ and the quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].To use the formula, follow these steps:

Identify the coefficients: In our equation, $a = 1$ , $b = 7.15$ , and $c = 3.24$.
Calculate the discriminant: Substitute the values into the discriminant formula to get $\Delta = 38.1625$.
Find the square root of the discriminant: $\sqrt{38.1625} \approx 6.177$ .
Apply the quadratic formula: Substitute the coefficients and the square root of the discriminant into the quadratic formula: $x = \frac{-7.15 \pm 6.177}{2} $.
Solve for the two values of $x$ : This will give us: $x_1 = \frac{-7.15 + 6.177}{2} = \frac{-0.973}{2} \approx -0.487$ and $x_2 = \frac{-7.15 - 6.177}{2} = \frac{-13.327}{2} \approx -6.664$.

Following these steps carefully ensures we accurately find the solutions of our quadratic equation.

Verification of Solutions

Once we have the solutions from the quadratic formula, it鈥檚 essential to verify if they are correct. To verify, substitute the solutions back into the original equation: $x^2 + 7.15x + 3.24 = 0$.
For $x_1 = -0.487$:

Calculate $(-0.487)^2 + 7.15(-0.487) + 3.24:$
$0.237 + (-3.481) + 3.24 \approx 0$.
This confirms $x_1$ is a valid solution.

For $x_2 = -6.664$:

Calculate $(-6.664)^2 + 7.15(-6.664) + 3.24:$
$44.415 + (-47.633) + 3.24 \approx 0$.
This confirms $x_2$ is a valid solution.

Verification helps ensure there are no calculation errors and that the solutions meet the original equation.

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91影视

Use the quadratic formula and a calculator to solve each equation. Round answers to three decimal places and check your answers. $$x^{2}+7.15 x+3.24=0$$

Short Answer

Step by step solution

Identify coefficients

Write the quadratic formula

Calculate the discriminant

Calculate the square root of the discriminant

Apply the quadratic formula

Solve for the two values of x

Verify the solutions

Key Concepts

Discriminant

Solving Quadratic Equations

Verification of Solutions

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