X 2 7x 8 0

$\exponential{(x)}{2} - seven x - 8 $

\left(x-8\right)\left(x+1\right)

\left(x-viii\right)\left(10+one\right)

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a+b=-7 ab=1\left(-viii\correct)=-8

Factor the expression past grouping. Starting time, the expression needs to be rewritten as x^{two}+ax+bx-eight. To observe a and b, ready upwards a system to be solved.

ane,-eight two,-4

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater accented value than the positive. List all such integer pairs that give production -8.

1-eight=-seven two-4=-2

Calculate the sum for each pair.

a=-eight b=1

The solution is the pair that gives sum -vii.

\left(x^{2}-8x\right)+\left(x-eight\right)

Rewrite ten^{2}-7x-8 every bit \left(x^{ii}-8x\right)+\left(x-8\correct).

x\left(10-8\correct)+x-8

Gene out 10 in ten^{2}-8x.

\left(10-8\correct)\left(ten+one\right)

Factor out common term 10-8 by using distributive property.

x^{ii}-7x-8=0

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(10-x_{2}\right), where x_{1} and x_{two} are the solutions of the quadratic equation ax^{2}+bx+c=0.

ten=\frac{-\left(-7\right)±\sqrt{\left(-7\correct)^{2}-4\left(-8\right)}}{2}

All equations of the course ax^{2}+bx+c=0 can exist solved using the quadratic formula: \frac{-b±\sqrt{b^{two}-4ac}}{2a}. The quadratic formula gives two solutions, ane when ± is improver and one when it is subtraction.

ten=\frac{-\left(-seven\right)±\sqrt{49-4\left(-eight\right)}}{2}

Square -seven.

x=\frac{-\left(-vii\right)±\sqrt{49+32}}{2}

Multiply -4 times -viii.

10=\frac{-\left(-vii\correct)±\sqrt{81}}{2}

Add 49 to 32.

x=\frac{-\left(-7\right)±9}{2}

Accept the foursquare root of 81.

x=\frac{7±ix}{two}

The opposite of -7 is seven.

ten=\frac{16}{two}

Now solve the equation x=\frac{seven±9}{2} when ± is plus. Add together 7 to 9.

x=\frac{-2}{2}

At present solve the equation x=\frac{7±9}{ii} when ± is minus. Decrease nine from vii.

x^{2}-7x-8=\left(10-8\right)\left(x-\left(-1\correct)\right)

Factor the original expression using ax^{ii}+bx+c=a\left(x-x_{1}\right)\left(10-x_{2}\correct). Substitute eight for x_{1} and -1 for x_{2}.

x^{2}-7x-8=\left(x-8\right)\left(x+1\right)

Simplify all the expressions of the form p-\left(-q\right) to p+q.

x ^ 2 -7x -8 = 0

Quadratic equations such as this one tin can be solved by a new direct factoring method that does not require estimate work. To use the direct factoring method, the equation must exist in the form x^2+Bx+C=0.

r + south = 7 rs = -8

Let r and southward be the factors for the quadratic equation such that x^two+Bx+C=(x−r)(x−due south) where sum of factors (r+due south)=−B and the product of factors rs = C

r = \frac{7}{2} - u due south = \frac{7}{2} + u

Two numbers r and s sum upwardly to 7 exactly when the average of the two numbers is \frac{i}{2}*7 = \frac{7}{2}. Yous can also encounter that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented past the quadratic equation y=x^two+Bx+C. The values of r and s are equidistant from the center past an unknown quantity u. Limited r and s with respect to variable u. <div mode='padding: 8px'><img src='https://opalmath.azureedge.cyberspace/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

(\frac{7}{2} - u) (\frac{vii}{two} + u) = -eight

To solve for unknown quantity u, substitute these in the product equation rs = -8

\frac{49}{4} - u^ii = -8

Simplify by expanding (a -b) (a + b) = a^2 – b^2

-u^ii = -8-\frac{49}{4} = -\frac{81}{four}

Simplify the expression past subtracting \frac{49}{4} on both sides

u^two = \frac{81}{4} u = \pm\sqrt{\frac{81}{4}} = \pm \frac{9}{2}

Simplify the expression past multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

r =\frac{vii}{2} - \frac{ix}{2} = -1 s = \frac{vii}{two} + \frac{nine}{2} = 8

The factors r and southward are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

X 2 7x 8 0,

Source: https://mathsolver.microsoft.com/en/solve-problem/%7B%20x%20%20%7D%5E%7B%202%20%20%7D%20%20-7x-8

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