Solve x^2-64x+348=0

$${ x }^{ 2 } -64x+348=0$$

Solve for x

$$x=6$$

$$x=58$$

Steps Using Factoring

To solve the equation, factor x^{2}-64x+348 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.

$$a+b=-64 ab=348$$

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 348.

$$-1,-348 -2,-174 -3,-116 -4,-87 -6,-58 -12,-29$$

Calculate the sum for each pair.

$$-1-348=-349 -2-174=-176 -3-116=-119 -4-87=-91 -6-58=-64 -12-29=-41$$

The solution is the pair that gives sum -64.

$$a=-58 b=-6$$

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

$$\left(x-58\right)\left(x-6\right)$$

To find equation solutions, solve x-58=0 and x-6=0.

$$x=58 x=6$$

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+348. To find a and b, set up a system to be solved.

$$a+b=-64 ab=1\times 348=348$$

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 348.

$$-1,-348 -2,-174 -3,-116 -4,-87 -6,-58 -12,-29$$

Calculate the sum for each pair.

$$-1-348=-349 -2-174=-176 -3-116=-119 -4-87=-91 -6-58=-64 -12-29=-41$$

The solution is the pair that gives sum -64.

$$a=-58 b=-6$$

Rewrite x^{2}-64x+348 as \left(x^{2}-58x\right)+\left(-6x+348\right).

$$\left(x^{2}-58x\right)+\left(-6x+348\right)$$

Factor out x in the first and -6 in the second group.

$$x\left(x-58\right)-6\left(x-58\right)$$

Factor out common term x-58 by using distributive property.

$$\left(x-58\right)\left(x-6\right)$$

To find equation solutions, solve x-58=0 and x-6=0.

$$x=58 x=6$$

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.

$$x^{2}-64x+348=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -64 for b, and 348 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-64\right)±\sqrt{\left(-64\right)^{2}-4\times 348}}{2}$$

Square -64.

$$x=\frac{-\left(-64\right)±\sqrt{4096-4\times 348}}{2}$$

Multiply -4 times 348.

$$x=\frac{-\left(-64\right)±\sqrt{4096-1392}}{2}$$

Add 4096 to -1392.

$$x=\frac{-\left(-64\right)±\sqrt{2704}}{2}$$

Take the square root of 2704.

$$x=\frac{-\left(-64\right)±52}{2}$$

The opposite of -64 is 64.

$$x=\frac{64±52}{2}$$

Now solve the equation x=\frac{64±52}{2} when ± is plus. Add 64 to 52.

$$x=\frac{116}{2}$$

Divide 116 by 2.

$$x=58$$

Now solve the equation x=\frac{64±52}{2} when ± is minus. Subtract 52 from 64.

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

The equation is now solved.

$$x=58 x=6$$

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.

$$x^{2}-64x+348=0$$

Subtract 348 from both sides of the equation.

$$x^{2}-64x+348-348=-348$$

Subtracting 348 from itself leaves 0.

$$x^{2}-64x=-348$$

Divide -64, the coefficient of the x term, by 2 to get -32. Then add the square of -32 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-64x+\left(-32\right)^{2}=-348+\left(-32\right)^{2}$$

Square -32.

$$x^{2}-64x+1024=-348+1024$$

Add -348 to 1024.

$$x^{2}-64x+1024=676$$

Factor x^{2}-64x+1024. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-32\right)^{2}=676$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-32\right)^{2}}=\sqrt{676}$$

Simplify.

$$x-32=26 x-32=-26$$

Add 32 to both sides of the equation.

$$x=58 x=6$$