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Chapter 8: Problem 16

Solve each equation by using the quadratic formula. $$4 x^{2}-12 x+9=0$$

Short Answer

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

Step by step solution

01

Identify coefficients

In a quadratic equation of the form \(ax^2 + bx + c = 0\), identify the coefficients \(a\), \(b\), and \(c\). For the equation \(4x^2 - 12x + 9 = 0\), we have \(a = 4\), \(b = -12\), and \(c = 9\).
02

Write down the quadratic formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula is used to find the solutions of any quadratic equation.
03

Substitute coefficients into the formula

Substitute \(a = 4\), \(b = -12\), and \(c = 9\) into the quadratic formula: \(x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \cdot 4 \cdot 9}}{2 \cdot 4}\).
04

Simplify inside the square root

Calculate the discriminant: \((-12)^2 - 4 \cdot 4 \cdot 9 = 144 - 144 = 0\). Now the equation becomes: \(x = \frac{12 \pm \sqrt{0}}{8}\).
05

Solve the equation

Since the discriminant is 0, there is only one solution: \(x = \frac{12}{8} = \frac{3}{2}\). Therefore, the solution to the equation \(4x^2 - 12x + 9 = 0\) is \(x = \frac{3}{2}\).

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Key Concepts

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

quadratic formula
The quadratic formula is a powerful tool that allows us to find the solutions to any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
To use this formula, you merely need to identify the coefficients \(a\), \(b\), and \(c\).
For example, in the equation \(4x^2 - 12x + 9 = 0\), the coefficients are:
  • \(a = 4\)
  • \(b = -12\)
  • \(c = 9\)
By substituting these values into the quadratic formula, you can solve for \(x\). This formula incorporates all the necessary calculations in one expression, making the process systematic and straightforward.
discriminant
The discriminant is a crucial part of the quadratic formula that helps us determine the nature of the roots of a quadratic equation. It is represented by:
\[\text{Discriminant} = b^2 - 4ac\]
The discriminant tells us how many solutions (or roots) the quadratic equation has and what type they are:
  • If the discriminant is positive (\( > 0 \)), there are two distinct real roots.
  • If the discriminant is zero (\( = 0 \)), there is exactly one real root.
  • If the discriminant is negative (\( < 0 \)), there are two complex roots.
In our example, \(4x^2 - 12x + 9 = 0\), the discriminant is:
\[(-12)^2 - 4 \times 4 \times 9 = 144 - 144 = 0\]
Since the discriminant is zero, the equation has one real root.
coefficients
Coefficients are the numerical constants that multiply the variables in an equation. In a quadratic equation of the form \(ax^2 + bx + c = 0\):
  • \(a\) is the coefficient of \(x^2\), which we call the quadratic coefficient.
  • \(b\) is the coefficient of \(x\), known as the linear coefficient.
  • \(c\) is the constant term, also known as the constant coefficient.
Accurately identifying these coefficients is the first step in solving quadratic equations using the quadratic formula.
For the equation \(4x^2 - 12x + 9 = 0\), we have:
  • Quadratic coefficient \(a = 4\)
  • Linear coefficient \(b = -12\)
  • Constant coefficient \(c = 9\)
By substituting these into the quadratic formula, we can find the solution efficiently.

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Most popular questions from this chapter

Use the quadratic formula and a calculator to solve each equation. Round answers to three decimal places and check your answers. $$1.85 x^{2}+6.72 x+3.6=0$$

Find the exact solution \((s)\) to each problem. If the solution(s) are irrational, then also find approximate solution(s) to the nearest tenth. Diagonal of a square. The diagonal of a square is \(2 \mathrm{m}\) longer than a side. Find the length of a side.

Cooperative learning. Work with a group to write a quadratic equation that has each given pair of solutions. a) \(3+\sqrt{5}, 3-\sqrt{5}\) b) \(4-2 i, 4+2 i\) c) \(\frac{1+i \sqrt{3}}{2}, \frac{1-i \sqrt{3}}{2}\)

Use the quadratic formula and a calculator to solve each equation. Round answers to three decimal places and check your answers. $$4.3 x^{2}-9.86 x-3.75=0$$

Find the solutions to \(6 x^{2}+5 x-4=0 .\) Is the sum of your solutions equal to \(-\frac{b}{a}\) ? Explain why the sum of the solutions to any quadratic equation is \(-\frac{b}{a}\) (Hint: Use the quadratic formula.)
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