Question #ab4db

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Question #ab4db

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Answer 1 · 2017-09-08T23:34:37

See explanation below.

Explanation:

Yes, you are right. The quadratic formula is one way to solve this particular equation. Alternatively, you can also solve by graphing. I won't go into too much detail on the actual process of solving this equation by graphing, but I will just say that you look where the graph intersects the $x$ $x$ -axis (these would be the solution(s)).

If we use the quadratic formula, we will have to associate the equation given with

$a x^{2} + b x + c = 0$ $ax^2 + bx + c = 0$ .

If we substitute our given equation $x^{2} - 10 x + 40 = 0$ $x^2-10x+40=0$ into the above equation, we get:

$a = 1$ $a=1$

$b = - 10$ $b=-10$

$c = 40$ $c=40$

These values give us just enough information to use the quadratic formula, which is:

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ $x = (-b +- sqrt(b^2 - 4ac))/(2a)$

So if we substitute our values that we found earlier into this unique formula, we will get $x$ $x$ .

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ $x = (-b +- sqrt(b^2 - 4ac))/(2a)$

$x = \frac{- (- 10) \pm \sqrt{{(- 10)}^{2} - 4 (1) (40)}}{2 (1)}$ $x = (-(-10) +- sqrt((-10)^2 - 4(1)(40)))/(2(1))$

$x = \frac{10 \pm \sqrt{100 - 160}}{2}$ $x=(10 +- sqrt(100-160))/(2)$

$x = \frac{10 \pm \sqrt{- 60}}{2}$ $x=(10+- sqrt(-60))/2$

At this point, you will see a negative square root, which indicates that the solution is no real numbers. But if you are in a higher level of math, and you have learned about imaginary numbers (like $i$ $i$ ), then you are not yet finished with the question. So, let's continue.

From here, we simplify the radical using the imaginary number $i$ $i$ ( $i = \sqrt{- 1}$ $i=sqrt(-1)$ ). See below.

$x = \frac{10 \pm \sqrt{- 60}}{2}$ $x=(10+- sqrt(-60))/2$

$x = \frac{10 \pm \sqrt{- 20 \cdot 2 \cdot 2}}{2}$ $x=(10 +- sqrt(-20 * 2 * 2))/(2)$

$x = \frac{10 \pm \sqrt{- 15 \cdot 2 \cdot 2}}{2}$ $x=(10 +- sqrt(-15 * 2 * 2))/(2)$

$x = \frac{10 \pm 2 \sqrt{- 15}}{2}$ $x=(10+-2sqrt(-15))/(2)$

$x = \frac{10 \pm 2 i \sqrt{15}}{2}$ $x=(10+-2isqrt(15))/(2)$

If you want an exact answer (no rounding), then this would be your answer. If you want to put the solutions in $a + b i$ $a+bi$ form, I leave the task to you, as we are not far off.

I hope that helps!

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Question #ab4db

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