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Chapter 1: Problem 23

Solve each equation by factoring. $$ 4 x^{2}+9=12 x $$

Short Answer

Expert verified
The solution is \(x = \frac{3}{2}\)

Step by step solution

01

Set the equation to zero

Rearrange the equation so that all terms are on one side. Move the term involving the variable to the left side of the equation: \[ 4x^{2} + 9 - 12x = 0 \]
02

Rewrite the equation

Rewrite the equation in standard quadratic form \[ax^{2} + bx + c = 0\]In this case, it becomes: \[ 4x^{2} - 12x + 9 = 0\]
03

Factor the quadratic equation

We need to find two binomials that multiply to give \[4x^{2} - 12x + 9\].Start by looking for two numbers that multiply to \[4 \times 9 = 36\] and add to \[-12\].The numbers \[-6\] and \[-6\] work.So, we rewrite the middle term and factor by grouping:\[4x^2 - 12x + 9 = (2x - 3)^2\]
04

Solve for x

Set each factor equal to zero:\[ 2x - 3 = 0 \]Solve for x:\[2x = 3\]\[x = \frac{3}{2}\]

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Key Concepts

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

Quadratic Equations
Quadratic equations are polynomial equations of the second degree. This means they have the form: \[ ax^2 + bx + c = 0 \] In this format, 'a', 'b', and 'c' are constants, and 'x' represents an unknown variable. The term that greatly distinguishes a quadratic equation is the 'ax^2' term, which contains an 'x' squared. Quadratic equations typically have two solutions, since they are derived from parabolic graphs that can cross the x-axis at up to two points.
Factoring
Factoring is a method used to break down an equation into simpler elements called factors that, when multiplied together, yield the original equation. In the quadratic equation, we rewrite it into the form of: \[ ax^2 + bx + c \] This helps identify two binomials that multiply together to produce the quadratic equation. For example, if the equation is: \[ 4x^2 - 12x + 9 \] we look for two numbers that multiply to \[ 4 \times 9 = 36 \] and add to -12. These numbers are -6 and -6. Thus, we rewrite the quadratic equation as: \[ (2x - 3)^2 \] Factoring simplifies solving the equation.
Solving Equations
After factoring the quadratic equation, the next step is to solve for 'x'. We do this by setting each factor equal to zero and solving for the variable. For the factored form \[ (2x - 3)^2 \] , we would set \[ 2x - 3 = 0 \] Solving this gives: \[ 2x = 3 \] Finally, divide both sides by 2: \[ x = \frac{3}{2} \] By solving each factor separately, we can find the value(s) of 'x' that satisfy the original equation. Remember that the quadratic equation can have two, one, or sometimes no real solutions depending on the discriminant.

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