Solve each quadratic equation. If roots are not real, use ii. w24+8=0\frac{w^2}{4}+8=0

w = ± 4 2 i w=\pm4\sqrt2i w = ± 4 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> i

Solve each quadratic equation. If roots are not real, use i i . w 2 4 + 8 = 0 \frac{w^2}{4}+8=0

Here, we're going to solve this quadratic equation using our square root property. So we want to isolate our variable w in order to find what value for w makes this equation true. So let's start by subtracting eight from both sides. So that we get w squared over four is equal to negative eight. Next, we'll multiply both sides by four. So that four over four cross out, and we have w squared is equal to negative eight times four, which is negative 32. Now that we have our quantity squared equal to some value, we can apply our square root property. So square rooting both sides, so that our square root and square cancel out. We get w is equal to positive and negative square root of negative 32. Lastly, we need to simplify our radical using I if needed. Using our product property, I know that I can split up this square root as the square root of 32 times the square root of negative one. And I know that the square root of negative one is I. So we can rewrite our answer as w is equal to positive negative square root of negative 32 times I. And lastly, let's simplify the square root of 32. Since it's an even number, I know it's divisible by two, and thirty two divided by two is 16. And 16 is a perfect square number with four and four. So we can pull out four and we get our final answer of w equals positive and negative four root two times I. And this here is our final answer.

Solve each quadratic equation. If roots are not real, use i i . w 2 4 + 8 = 0 \frac{w^2}{4}+8=0

w = ± 4 2 i w=\pm4\sqrt2i w = ± 4 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> i

w = ± 4 2 w=\pm4\sqrt2 w = ± 4 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

10. Quadratic Equations and Functions

The Square Root Property

Intermediate Algebra

10. Quadratic Equations and Functions

The Square Root Property

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Multiple Choice

Solve each quadratic equation. If roots are not real, use $i$ .
$\frac{w^{2}}{4} + 8 = 0$

$w=\pm4\sqrt2i$

$w=\pm4i$

$w=\pm4$

$w=\pm4\sqrt2$

Verified step by step guidance

Start with the given equation: $\frac{w^2}{4} + 8 = 0$.

Isolate the term with $w^2$ by subtracting 8 from both sides: $\frac{w^2}{4} = -8$.

Eliminate the denominator by multiplying both sides of the equation by 4: $w^2 = -8 \times 4$.

Simplify the right side to get $w^2 = -32$.

Since $w^2$ equals a negative number, take the square root of both sides and include the imaginary unit $i$ to represent the square root of a negative number: $w = \pm \sqrt{-32} = \pm \sqrt{32}i$.

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