Solve 4x^2-45=31x

$$4 { x }^{ 2 } -45=31x$$

Solve for x

$$x = -\frac{5}{4} = -1\frac{1}{4} = -1.25$$

$$x=9$$

Steps Using Factoring By Grouping

Subtract 31x from both sides.

$$4x^{2}-45-31x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$4x^{2}-31x-45=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4x^{2}+ax+bx-45. To find a and b, set up a system to be solved.

$$a+b=-31 ab=4\left(-45\right)=-180$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -180.

$$1,-180 2,-90 3,-60 4,-45 5,-36 6,-30 9,-20 10,-18 12,-15$$

Calculate the sum for each pair.

$$1-180=-179 2-90=-88 3-60=-57 4-45=-41 5-36=-31 6-30=-24 9-20=-11 10-18=-8 12-15=-3$$

The solution is the pair that gives sum -31.

$$a=-36 b=5$$

Rewrite 4x^{2}-31x-45 as \left(4x^{2}-36x\right)+\left(5x-45\right).

$$\left(4x^{2}-36x\right)+\left(5x-45\right)$$

Factor out 4x in the first and 5 in the second group.

$$4x\left(x-9\right)+5\left(x-9\right)$$

Factor out common term x-9 by using distributive property.

$$\left(x-9\right)\left(4x+5\right)$$

To find equation solutions, solve x-9=0 and 4x+5=0.

$$x=9 x=-\frac{5}{4}$$

Steps Using the Quadratic Formula

Subtract 31x from both sides.

$$4x^{2}-45-31x=0$$

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.

$$4x^{2}-31x-45=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -31 for b, and -45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-31\right)±\sqrt{\left(-31\right)^{2}-4\times 4\left(-45\right)}}{2\times 4}$$

Square -31.

$$x=\frac{-\left(-31\right)±\sqrt{961-4\times 4\left(-45\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-31\right)±\sqrt{961-16\left(-45\right)}}{2\times 4}$$

Multiply -16 times -45.

$$x=\frac{-\left(-31\right)±\sqrt{961+720}}{2\times 4}$$

Add 961 to 720.

$$x=\frac{-\left(-31\right)±\sqrt{1681}}{2\times 4}$$

Take the square root of 1681.

$$x=\frac{-\left(-31\right)±41}{2\times 4}$$

The opposite of -31 is 31.

$$x=\frac{31±41}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{31±41}{8}$$

Now solve the equation x=\frac{31±41}{8} when ± is plus. Add 31 to 41.

$$x=\frac{72}{8}$$

Divide 72 by 8.

$$x=9$$

Now solve the equation x=\frac{31±41}{8} when ± is minus. Subtract 41 from 31.

$$x=-\frac{10}{8}$$

Reduce the fraction \frac{-10}{8} to lowest terms by extracting and canceling out 2.

$$x=-\frac{5}{4}$$

The equation is now solved.

$$x=9 x=-\frac{5}{4}$$

Steps for Completing the Square

Subtract 31x from both sides.

$$4x^{2}-45-31x=0$$

Add 45 to both sides. Anything plus zero gives itself.

$$4x^{2}-31x=45$$

Divide both sides by 4.

$$\frac{4x^{2}-31x}{4}=\frac{45}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}-\frac{31}{4}x=\frac{45}{4}$$

Divide -\frac{31}{4}, the coefficient of the x term, by 2 to get -\frac{31}{8}. Then add the square of -\frac{31}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{31}{4}x+\left(-\frac{31}{8}\right)^{2}=\frac{45}{4}+\left(-\frac{31}{8}\right)^{2}$$

Square -\frac{31}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{31}{4}x+\frac{961}{64}=\frac{45}{4}+\frac{961}{64}$$

Add \frac{45}{4} to \frac{961}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{31}{4}x+\frac{961}{64}=\frac{1681}{64}$$

Factor x^{2}-\frac{31}{4}x+\frac{961}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{31}{8}\right)^{2}=\frac{1681}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{31}{8}\right)^{2}}=\sqrt{\frac{1681}{64}}$$

Simplify.

$$x-\frac{31}{8}=\frac{41}{8} x-\frac{31}{8}=-\frac{41}{8}$$

Add \frac{31}{8} to both sides of the equation.

$$x=9 x=-\frac{5}{4}$$