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Chapter 0: Problem 81

Solve each quadratic equation using the quadratic formula. $$4 x^{2}=2 x+7$$

Short Answer

Expert verified
The solution to the equation is \(x \approx 1.34\) and \(x \approx -1.34\)

Step by step solution

01

Plug the coefficients into the quadratic formula.

The quadratic formula is \(x =\frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). Replace a, b, and c with the coefficients from the equation. Thus we get, \(x =\frac{2 \pm \sqrt{(-2)^{2} - 4*4*(-7)}}{2 * 4}\)
02

Simplify the expression under the square root (the discriminant).

Now we will calculate the expression under the square root, which is called the discriminant. The discriminant is \((-2)^{2} - 4*4*(-7)\) = 4 + 112 = 116. Our equation now looks like this: \(x =\frac{2 \pm \sqrt{116}}{8}\)
03

Calculate the roots

Now we will calculate the roots using the plus-minus part of the quadratic formula. This gives us the two possible solutions for x:1) When using the +, \(x = \frac{2 + \sqrt{116}}{8} \approx 1.34\)2) When using the -, \(x = \frac{2 - \sqrt{116}}{8} \approx -1.34\)

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Most popular questions from this chapter

Rationalize the numerator. $$\frac{\sqrt{x}+\sqrt{y}}{x^{2}-y^{2}}$$

Perform the indicated operations. Simplify the result, if possible. $$\left(4-\frac{3}{x+2}\right)\left(1+\frac{5}{x-1}\right)$$

Place the correct symbol, \(>\) or \(<\), in the shaded area between the given numbers. Do not use a calculator. Then check your result with a calculator. \(a, 3^{\frac{1}{2}}\square 3^{1}\) b. \(\sqrt{7}+\sqrt{18} \square \sqrt{7+18}\)

Explain how to determine which numbers must be excluded from the domain of a rational expression.

Perform the indicated operations. Simplify the result, if possible. $$\frac{y^{-1}-(y+5)^{-1}}{5}$$
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