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Chapter 5: Problem 30

Solve each equation, and check the solutions. $$ x^{2}=4+3 x $$

Short Answer

Expert verified
The solutions are \( x = 4 \) and \( x = -1 \).

Step by step solution

01

- Rewrite the Equation

Rewrite the given equation in standard form as a quadratic equation, bringing all terms to one side: \[ x^2 - 3x - 4 = 0 \]
02

- Factor the Quadratic Equation

Factor the quadratic equation. We are looking for two numbers that multiply to -4 and add to -3: \[ (x - 4)(x + 1) = 0 \]
03

- Solve for x

Set each factor to zero and solve for x: \[ x - 4 = 0 \] \[ x = 4 \] \[ x + 1 = 0 \] \[ x = -1 \]
04

- Check the Solutions

Substitute each solution back into the original equation to check if they satisfy it: For \( x = 4 \): \[ 4^2 = 4 + 3(4) \] \[ 16 = 4 + 12 \] \[ 16 = 16 \] For \( x = -1 \): \[ (-1)^2 = 4 + 3(-1) \] \[ 1 = 4 - 3 \] \[ 1 = 1 \]

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Key Concepts

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

Factoring Quadratic Equations
Factoring is an essential algebra skill used to solve quadratic equations. It transforms a quadratic equation into a product of two binomials. Here, you need to find two numbers that multiply to a constant term (in this case, -4) and add up to the linear coefficient (-3). Using the equation \(x^2 - 3x - 4 = 0\), the factored form becomes \( (x - 4)(x + 1) = 0\). By setting each binomial to zero, the original quadratic equation can be split into simpler linear equations.
Checking Solutions in Algebra
Verifying your solutions ensures they satisfy the original equation. Once you solve for \(x = 4\) and \(x = -1\), check these solutions by substituting them back into the original equation. For \( x = 4 \), substituting in \(x^2 = 4 + 3x\) gives \[16 = 4 + 12 \]. Similarly, for \( x = -1 \), it confirms \[1 = 4 - 3 \]. Both substitutions check out, confirming that the solutions are correct. This step is critical to avoid calculation errors and ensure the right answers.
Quadratic Equation Standard Form
The standard form of a quadratic equation is \[ax^2 + bx + c = 0 \]. In your given problem, you first need to rearrange \[x^2 = 4 + 3x \] into this standard form. By subtracting \[4 + 3x\], you end up with \[x^2 - 3x - 4 = 0 \]. Rewriting the equation this way is crucial as it sets the stage for methods like factoring, making it easier to solve the equation.
Substituting Solutions
Substitution involves replacing the variable with your found solutions. This step confirms that each solution works with the original equation. Take \(x = 4\) and substitute it back to verify: \[4^2 = 4 + 3(4) \]. Similarly, for \( x = -1 \): \[(-1)^2 = 4 + 3(-1) \]. Successful substitution where both sides of the equation match shows that the solutions are accurate. It is a reliable method to ensure your answers are correct and gives you confidence in your solutions.

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Most popular questions from this chapter

Solve each problem. Find three consecutive even integers such that the sum of the squares of the first and second integers is equal to the square of the third integer.

Solve each problem. Find two consecutive odd integers such that their product is 15 more than three times their sum.

Factor each binomial completely. \(27 t^{3}-64 s^{6}\)

Work each problem. To help factor sums or differences of cubes, complete the following list of perfect cubes. \(1^{3}=\)____ \(6^{3}=\)____ \(2^{3}=\)____ \(7^{3}=\)____ \(3^{3}=\)____ \(8^{3}=\)____ \(4^{3}=\)____ \(9^{3}=\)____ \(10^{3}=\)____

Solve each problem. The product of the second and third of three consecutive integers is 2 more than 10 times the first integer. Find the integers.
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