Question

Consider this inequality: 8x^2+ 2x – 3 ≥ 0. The quadratic formula gives the x-values that result in the quadratic expression being equal to 0. However, you can use these results to factor the quadratic inequality when it is difficult to factor and determine intervals to test when solving a quadratic inequality. To find an equivalent factored expression, subtract each result from x to get two factors. Then multiply the result by the leading coefficient, which in this case is 8. Apply the quadratic formula to determine which values of x will result in 0. Show your work.

Answer 1

Given quadratic 8x^2+ 2x – 3 ≥ 0.

We know, quadratic formula.

$\quad x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

For given quadratic values of a, b and c are

$a=8,\:b=2,\:c=-3$

Plugging values in quadratic formula, we get

$x_{1,\:2}=\frac{-2\pm \sqrt{2^2-4\cdot \:8\left(-3\right)}}{2\cdot \:8}$

$x=\frac{-2+\sqrt{2^2-4\cdot \:8\left(-3\right)}}{2\cdot \:8}$

$=\frac{-2+10}{16}$

$=\frac{1}{2}$

$x=\frac{-2-\sqrt{2^2-4\cdot \:8\left(-3\right)}}{2\cdot \:8}$

$=\frac{-2-\sqrt{100}}{16}$

$=\frac{-12}{16}$

$=-\frac{3}{4}$

$x=\frac{1}{2},\:x=-\frac{3}{4}$

Multiplying both sides by in $x=\frac{1}{2}$, we get

$2x=1$

Subtacting 1 from both sides, we get

2x-1=0

Multiplying by 4 on both sides in $x=-\frac{3}{4}$, we get

$4x=-3$

Adding 3 on both sides , we get

$4x+3=0$

Therefore, we got factors

(2x-1) (4x+3)=0

So, x=1/2 and x=-3/4 would result in 0.