Solve x^2-3x+53=3x+44

$$x ^ { 2 } - 3 x + 53 = 3 x + 44$$

Solve for x

$$x=3$$

Steps Using Factoring

Subtract 3x from both sides.

$$x^{2}-3x+53-3x=44$$

Combine -3x and -3x to get -6x.

$$x^{2}-6x+53=44$$

Subtract 44 from both sides.

$$x^{2}-6x+53-44=0$$

Subtract 44 from 53 to get 9.

$$x^{2}-6x+9=0$$

To solve the equation, factor x^{2}-6x+9 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.

$$a+b=-6 ab=9$$

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 9.

$$-1,-9 -3,-3$$

Calculate the sum for each pair.

$$-1-9=-10 -3-3=-6$$

The solution is the pair that gives sum -6.

$$a=-3 b=-3$$

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

$$\left(x-3\right)\left(x-3\right)$$

Rewrite as a binomial square.

$$\left(x-3\right)^{2}$$

To find equation solution, solve x-3=0.

$$x=3$$

Steps Using Factoring By Grouping

Subtract 3x from both sides.

$$x^{2}-3x+53-3x=44$$

Combine -3x and -3x to get -6x.

$$x^{2}-6x+53=44$$

Subtract 44 from both sides.

$$x^{2}-6x+53-44=0$$

Subtract 44 from 53 to get 9.

$$x^{2}-6x+9=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+9. To find a and b, set up a system to be solved.

$$a+b=-6 ab=1\times 9=9$$

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 9.

$$-1,-9 -3,-3$$

Calculate the sum for each pair.

$$-1-9=-10 -3-3=-6$$

The solution is the pair that gives sum -6.

$$a=-3 b=-3$$

Rewrite x^{2}-6x+9 as \left(x^{2}-3x\right)+\left(-3x+9\right).

$$\left(x^{2}-3x\right)+\left(-3x+9\right)$$

Factor out x in the first and -3 in the second group.

$$x\left(x-3\right)-3\left(x-3\right)$$

Factor out common term x-3 by using distributive property.

$$\left(x-3\right)\left(x-3\right)$$

Rewrite as a binomial square.

$$\left(x-3\right)^{2}$$

To find equation solution, solve x-3=0.

$$x=3$$

Steps Using the Quadratic Formula

Subtract 3x from both sides.

$$x^{2}-3x+53-3x=44$$

Combine -3x and -3x to get -6x.

$$x^{2}-6x+53=44$$

Subtract 44 from both sides.

$$x^{2}-6x+53-44=0$$

Subtract 44 from 53 to get 9.

$$x^{2}-6x+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 9}}{2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 9}}{2}$$

Multiply -4 times 9.

$$x=\frac{-\left(-6\right)±\sqrt{36-36}}{2}$$

Add 36 to -36.

$$x=\frac{-\left(-6\right)±\sqrt{0}}{2}$$

Take the square root of 0.

$$x=-\frac{-6}{2}$$

The opposite of -6 is 6.

$$x=\frac{6}{2}$$

Divide 6 by 2.

$$x=3$$

Steps for Completing the Square

Subtract 3x from both sides.

$$x^{2}-3x+53-3x=44$$

Combine -3x and -3x to get -6x.

$$x^{2}-6x+53=44$$

Subtract 53 from both sides.

$$x^{2}-6x=44-53$$

Subtract 53 from 44 to get -9.

$$x^{2}-6x=-9$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=-9+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-9+9$$

Add -9 to 9.

$$x^{2}-6x+9=0$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-3=0 x-3=0$$

Add 3 to both sides of the equation.

$$x=3 x=3$$

The equation is now solved. Solutions are the same.

$$x=3$$