Solve x^2-25x+144=0quad15

$$x ^ { 2 } - 25 x + 144 = 0 \quad 15$$

Solve for x

$$x=9$$

$$x=16$$

Steps Using Factoring

Multiply 0 and 15 to get 0.

$$x^{2}-25x+144=0$$

To solve the equation, factor x^{2}-25x+144 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.

$$a+b=-25 ab=144$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 144.

$$-1,-144 -2,-72 -3,-48 -4,-36 -6,-24 -8,-18 -9,-16 -12,-12$$

Calculate the sum for each pair.

$$-1-144=-145 -2-72=-74 -3-48=-51 -4-36=-40 -6-24=-30 -8-18=-26 -9-16=-25 -12-12=-24$$

The solution is the pair that gives sum -25.

$$a=-16 b=-9$$

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

$$\left(x-16\right)\left(x-9\right)$$

To find equation solutions, solve x-16=0 and x-9=0.

$$x=16 x=9$$

Steps Using Factoring By Grouping

Multiply 0 and 15 to get 0.

$$x^{2}-25x+144=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+144. To find a and b, set up a system to be solved.

$$a+b=-25 ab=1\times 144=144$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 144.

$$-1,-144 -2,-72 -3,-48 -4,-36 -6,-24 -8,-18 -9,-16 -12,-12$$

Calculate the sum for each pair.

$$-1-144=-145 -2-72=-74 -3-48=-51 -4-36=-40 -6-24=-30 -8-18=-26 -9-16=-25 -12-12=-24$$

The solution is the pair that gives sum -25.

$$a=-16 b=-9$$

Rewrite x^{2}-25x+144 as \left(x^{2}-16x\right)+\left(-9x+144\right).

$$\left(x^{2}-16x\right)+\left(-9x+144\right)$$

Factor out x in the first and -9 in the second group.

$$x\left(x-16\right)-9\left(x-16\right)$$

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

$$\left(x-16\right)\left(x-9\right)$$

To find equation solutions, solve x-16=0 and x-9=0.

$$x=16 x=9$$

Steps Using the Quadratic Formula

Multiply 0 and 15 to get 0.

$$x^{2}-25x+144=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -25 for b, and 144 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 144}}{2}$$

Square -25.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\times 144}}{2}$$

Multiply -4 times 144.

$$x=\frac{-\left(-25\right)±\sqrt{625-576}}{2}$$

Add 625 to -576.

$$x=\frac{-\left(-25\right)±\sqrt{49}}{2}$$

Take the square root of 49.

$$x=\frac{-\left(-25\right)±7}{2}$$

The opposite of -25 is 25.

$$x=\frac{25±7}{2}$$

Now solve the equation x=\frac{25±7}{2} when ± is plus. Add 25 to 7.

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

Divide 32 by 2.

$$x=16$$

Now solve the equation x=\frac{25±7}{2} when ± is minus. Subtract 7 from 25.

$$x=\frac{18}{2}$$

Divide 18 by 2.

$$x=9$$

The equation is now solved.

$$x=16 x=9$$

Steps for Completing the Square

Multiply 0 and 15 to get 0.

$$x^{2}-25x+144=0$$

Subtract 144 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-25x=-144$$

Divide -25, the coefficient of the x term, by 2 to get -\frac{25}{2}. Then add the square of -\frac{25}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-25x+\left(-\frac{25}{2}\right)^{2}=-144+\left(-\frac{25}{2}\right)^{2}$$

Square -\frac{25}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-25x+\frac{625}{4}=-144+\frac{625}{4}$$

Add -144 to \frac{625}{4}.

$$x^{2}-25x+\frac{625}{4}=\frac{49}{4}$$

Factor x^{2}-25x+\frac{625}{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}.

$$\left(x-\frac{25}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$x-\frac{25}{2}=\frac{7}{2} x-\frac{25}{2}=-\frac{7}{2}$$

Add \frac{25}{2} to both sides of the equation.

$$x=16 x=9$$