Solve x^2+65x-1296=0

$${ x }^{ 2 } +65x-1296=0$$

Solve for x

$$x=-81$$

$$x=16$$

Steps Using Factoring

To solve the equation, factor x^{2}+65x-1296 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.

$$a+b=65 ab=-1296$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -1296.

$$-1,1296 -2,648 -3,432 -4,324 -6,216 -8,162 -9,144 -12,108 -16,81 -18,72 -24,54 -27,48 -36,36$$

Calculate the sum for each pair.

$$-1+1296=1295 -2+648=646 -3+432=429 -4+324=320 -6+216=210 -8+162=154 -9+144=135 -12+108=96 -16+81=65 -18+72=54 -24+54=30 -27+48=21 -36+36=0$$

The solution is the pair that gives sum 65.

$$a=-16 b=81$$

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

$$\left(x-16\right)\left(x+81\right)$$

To find equation solutions, solve x-16=0 and x+81=0.

$$x=16 x=-81$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-1296. To find a and b, set up a system to be solved.

$$a+b=65 ab=1\left(-1296\right)=-1296$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -1296.

$$-1,1296 -2,648 -3,432 -4,324 -6,216 -8,162 -9,144 -12,108 -16,81 -18,72 -24,54 -27,48 -36,36$$

Calculate the sum for each pair.

$$-1+1296=1295 -2+648=646 -3+432=429 -4+324=320 -6+216=210 -8+162=154 -9+144=135 -12+108=96 -16+81=65 -18+72=54 -24+54=30 -27+48=21 -36+36=0$$

The solution is the pair that gives sum 65.

$$a=-16 b=81$$

Rewrite x^{2}+65x-1296 as \left(x^{2}-16x\right)+\left(81x-1296\right).

$$\left(x^{2}-16x\right)+\left(81x-1296\right)$$

Factor out x in the first and 81 in the second group.

$$x\left(x-16\right)+81\left(x-16\right)$$

Factor out common term x-16 by using distributive property.

$$\left(x-16\right)\left(x+81\right)$$

To find equation solutions, solve x-16=0 and x+81=0.

$$x=16 x=-81$$

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.

$$x^{2}+65x-1296=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 65 for b, and -1296 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-65±\sqrt{65^{2}-4\left(-1296\right)}}{2}$$

Square 65.

$$x=\frac{-65±\sqrt{4225-4\left(-1296\right)}}{2}$$

Multiply -4 times -1296.

$$x=\frac{-65±\sqrt{4225+5184}}{2}$$

Add 4225 to 5184.

$$x=\frac{-65±\sqrt{9409}}{2}$$

Take the square root of 9409.

$$x=\frac{-65±97}{2}$$

Now solve the equation x=\frac{-65±97}{2} when ± is plus. Add -65 to 97.

$$x=\frac{32}{2}$$

Divide 32 by 2.

$$x=16$$

Now solve the equation x=\frac{-65±97}{2} when ± is minus. Subtract 97 from -65.

$$x=-\frac{162}{2}$$

Divide -162 by 2.

$$x=-81$$

The equation is now solved.

$$x=16 x=-81$$

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.

$$x^{2}+65x-1296=0$$

Add 1296 to both sides of the equation.

$$x^{2}+65x-1296-\left(-1296\right)=-\left(-1296\right)$$

Subtracting -1296 from itself leaves 0.

$$x^{2}+65x=-\left(-1296\right)$$

Subtract -1296 from 0.

$$x^{2}+65x=1296$$

Divide 65, the coefficient of the x term, by 2 to get \frac{65}{2}. Then add the square of \frac{65}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+65x+\left(\frac{65}{2}\right)^{2}=1296+\left(\frac{65}{2}\right)^{2}$$

Square \frac{65}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+65x+\frac{4225}{4}=1296+\frac{4225}{4}$$

Add 1296 to \frac{4225}{4}.

$$x^{2}+65x+\frac{4225}{4}=\frac{9409}{4}$$

Factor x^{2}+65x+\frac{4225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{65}{2}\right)^{2}=\frac{9409}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{65}{2}\right)^{2}}=\sqrt{\frac{9409}{4}}$$

Simplify.

$$x+\frac{65}{2}=\frac{97}{2} x+\frac{65}{2}=-\frac{97}{2}$$

Subtract \frac{65}{2} from both sides of the equation.

$$x=16 x=-81$$