How do you solve $y = 5 x^{2} + 20 x + 23$ $y=5x^2+20x+23$ using the completing square method?

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How do you solve $y = 5 x^{2} + 20 x + 23$ $y=5x^2+20x+23$ using the completing square method?

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Answer 1 · 2016-12-27T08:40:38

$x = - 2 \pm \sqrt{\frac{y - 3}{5}}$ $x = -2+-sqrt((y-3)/5)$

Explanation:

I will assume that we want to solve the equation in the sense of expressing possible values of $x$ $x$ in terms of $y$ $y$ .

If you actually just wanted to know the zeros, we can then put $y = 0$ $y=0$ and evaluate the result.

Given:

$y = 5 x^{2} + 20 x + 23$ $y = 5x^2+20x+23$

We will perform a sequence of steps to isolate $x$ $x$ on one side.

First divide both sides by $5$ $5$ to make the coefficient of $x^{2}$ $x^2$ into $1$ $1$ :

$\frac{y}{5} = x^{2} + 4 x + \frac{23}{5}$ $y/5 = x^2+4x+23/5$

If we take half of the coefficient of $x$ $x$ then we get the value $2$ $2$ , so note that:

${(x + 2)}^{2} = x^{2} + 4 x + 4$ $(x+2)^2 = x^2+4x+4$

which matches the right hand side of our equation in the first two terms.

So we can proceed as follows:

$\frac{y}{5} = x^{2} + 4 x + \frac{23}{5}$ $y/5 = x^2+4x+23/5$

$\frac{y}{5} = x^{2} + 4 x + 4 - 4 + \frac{23}{5}$ $color(white)(y/5) = x^2+4x+4-4+23/5$

$\frac{y}{5} = {(x + 2)}^{2} + \frac{3}{5}$ $color(white)(y/5) = (x+2)^2+3/5$

Subtracting $\frac{3}{5}$ $3/5$ from both ends we get:

$\frac{y - 3}{5} = {(x + 2)}^{2}$ $(y-3)/5 = (x+2)^2$

Transposed:

${(x + 2)}^{2} = \frac{y - 3}{5}$ $(x+2)^2 = (y-3)/5$

Take the square root of both sides, allowing for the possibility of either sign of square root to get:

$x + 2 = \pm \sqrt{\frac{y - 3}{5}}$ $x+2 = +-sqrt((y-3)/5)$

Finally subtract $2$ $2$ from both sides to find:

$x = - 2 \pm \sqrt{\frac{y - 3}{5}}$ $x = -2+-sqrt((y-3)/5)$

If we put $y = 0$ $y=0$ to find the zero of the quadratic then we get a pair of non-Real Complex values:

$x = - 2 \pm \sqrt{\frac{0 - 3}{5}} = - 2 \pm \sqrt{- \frac{3}{5}} = - 2 \pm \sqrt{\frac{3}{5}} i = - 2 \pm \frac{\sqrt{15}}{5} i$ $x = -2+-sqrt((0-3)/5) = -2+-sqrt(-3/5) = -2+-sqrt(3/5)i = -2+-sqrt(15)/5i$

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How do you solve $y = 5 x^{2} + 20 x + 23$ $y=5x^2+20x+23$ using the completing square method?

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How do you solve y=5x^2+20x+23y=5x2+20x+23y=5x^2+20x+23 using the completing square method?

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Explanation:

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How do you solve $y = 5 x^{2} + 20 x + 23$ $y=5x^2+20x+23$ using the completing square method?