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Chapter 5: Problem 27

Solve each equation, and check the solutions. $$ x^{2}=24-5 x $$

Short Answer

Expert verified
The solutions are x = -8 and x = 3.

Step by step solution

01

Set the equation to zero

First, rearrange the given equation so that one side equals zero. Start with the equation \[ x^2 = 24 - 5x \]Subtract 24 and add 5x to both sides to get:\[ x^2 + 5x - 24 = 0 \]
02

Factor the quadratic equation

To solve for x, factor the quadratic equation. Look for two numbers that multiply to -24 and add to 5. These numbers are 8 and -3, so the equation can be factored as:\[ (x + 8)(x - 3) = 0 \]
03

Solve for x

Set each factor equal to zero and solve for x:\[ x + 8 = 0 \] \[ x = -8 \] and\[ x - 3 = 0 \] \[ x = 3 \]
04

Verify the solutions

Substitute the solutions back into the original equation to check. For x = -8:\[ (-8)^2 = 24 - 5(-8) \]\[ 64 = 24 + 40 \]\[ 64 = 64 \] For x = 3:\[ 3^2 = 24 - 5(3) \]\[ 9 = 24 - 15 \]\[ 9 = 9 \]Both solutions satisfy the original equation.

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Key Concepts

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

factoring quadratics
Solving quadratic equations often involves factoring. Consider the quadratic equation in the form \[Ax^2 + Bx + C = 0\]. Factoring means rewriting this equation as a product of two binomials.

For example, let's factor the quadratic expression \[x^2 + 5x - 24\]. We need to find two numbers that multiply to \(-24\) and add up to \(5\). These numbers are \(8\) and \(-3\), so the equation can be written as \[(x + 8)(x - 3) = 0\].

Factoring is a crucial step because it transforms the quadratic into a simpler form, allowing us to solve for \(x\) by setting each binomial equal to zero.
verifying solutions
After finding potential solutions to a quadratic equation by factoring, it鈥檚 important to verify these solutions. This ensures that they satisfy the original equation.

Take our solutions \(x = -8\) and \(x = 3\) from the equation \[x^2 + 5x - 24 = 0\]. Substituting \(x = -8\) back into the original equation \[x^2 = 24 - 5x\]: \[ (-8)^2 = 24 - 5(-8) \]\[ 64 = 24 + 40 \]\[ 64 = 64 \].

Now, test \(x = 3\): \[ 3^2 = 24 - 5(3) \]\[ 9 = 24 - 15 \]\[ 9 = 9 \].

Since both values satisfy the original equation, the solutions are verified.
quadratic equations
Quadratic equations are polynomial equations of the form \[Ax^2 + Bx + C = 0\], where \(A, B\), and \(C\) are constants, and \(A 鈮� 0\). These equations are fundamental in algebra and arise in various real-world contexts.

The standard methods for solving quadratic equations include factoring, using the quadratic formula, and completing the square. In our example, we solved the equation \[x^2 = 24 - 5x\] through factoring: \[x^2 + 5x - 24 = 0 \] \[(x + 8)(x - 3) = 0\].

These equations often have two solutions, as seen in our case. Learning to handle quadratics is essential for progressing through more advanced mathematics.

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