Solve x^2-20x+96=77

$${ x }^{ 2 } -20x+96 = 77$$

Solve for x

$$x=1$$

$$x=19$$

Steps Using Factoring

Subtract 77 from both sides.

$$x^{2}-20x+96-77=0$$

Subtract 77 from 96 to get 19.

$$x^{2}-20x+19=0$$

To solve the equation, factor x^{2}-20x+19 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-20 ab=19$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-19 b=-1$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-19\right)\left(x-1\right)$$

To find equation solutions, solve x-19=0 and x-1=0.

$$x=19 x=1$$

Steps Using Factoring By Grouping

Subtract 77 from both sides.

$$x^{2}-20x+96-77=0$$

Subtract 77 from 96 to get 19.

$$x^{2}-20x+19=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+19. To find a and b, set up a system to be solved.

$$a+b=-20 ab=1\times 19=19$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-19 b=-1$$

Rewrite x^{2}-20x+19 as \left(x^{2}-19x\right)+\left(-x+19\right).

$$\left(x^{2}-19x\right)+\left(-x+19\right)$$

Factor out x in the first and -1 in the second group.

$$x\left(x-19\right)-\left(x-19\right)$$

Factor out common term x-19 by using distributive property.

$$\left(x-19\right)\left(x-1\right)$$

To find equation solutions, solve x-19=0 and x-1=0.

$$x=19 x=1$$

Steps Using the Quadratic Formula

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

$$x^{2}-20x+96=77$$

Subtract 77 from both sides of the equation.

$$x^{2}-20x+96-77=77-77$$

Subtracting 77 from itself leaves 0.

$$x^{2}-20x+96-77=0$$

Subtract 77 from 96.

$$x^{2}-20x+19=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -20 for b, and 19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 19}}{2}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\times 19}}{2}$$

Multiply -4 times 19.

$$x=\frac{-\left(-20\right)±\sqrt{400-76}}{2}$$

Add 400 to -76.

$$x=\frac{-\left(-20\right)±\sqrt{324}}{2}$$

Take the square root of 324.

$$x=\frac{-\left(-20\right)±18}{2}$$

The opposite of -20 is 20.

$$x=\frac{20±18}{2}$$

Now solve the equation x=\frac{20±18}{2} when ± is plus. Add 20 to 18.

$$x=\frac{38}{2}$$

Divide 38 by 2.

$$x=19$$

Now solve the equation x=\frac{20±18}{2} when ± is minus. Subtract 18 from 20.

$$x=\frac{2}{2}$$

Divide 2 by 2.

$$x=1$$

The equation is now solved.

$$x=19 x=1$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}-20x+96=77$$

Subtract 96 from both sides of the equation.

$$x^{2}-20x+96-96=77-96$$

Subtracting 96 from itself leaves 0.

$$x^{2}-20x=77-96$$

Subtract 96 from 77.

$$x^{2}-20x=-19$$

Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-20x+\left(-10\right)^{2}=-19+\left(-10\right)^{2}$$

Square -10.

$$x^{2}-20x+100=-19+100$$

Add -19 to 100.

$$x^{2}-20x+100=81$$

Factor x^{2}-20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-10\right)^{2}=81$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-10\right)^{2}}=\sqrt{81}$$

Simplify.

$$x-10=9 x-10=-9$$

Add 10 to both sides of the equation.

$$x=19 x=1$$