Solve 4x^2-90x+464

$$4 { x }^{ 2 } -90x+464$$

Factor

$$2\left(x-8\right)\left(2x-29\right)$$

Solution Steps

Factor out 2.

$$2\left(2x^{2}-45x+232\right)$$

Consider 2x^{2}-45x+232. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+232. To find a and b, set up a system to be solved.

$$a+b=-45 ab=2\times 232=464$$

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 464.

$$-1,-464 -2,-232 -4,-116 -8,-58 -16,-29$$

Calculate the sum for each pair.

$$-1-464=-465 -2-232=-234 -4-116=-120 -8-58=-66 -16-29=-45$$

The solution is the pair that gives sum -45.

$$a=-29 b=-16$$

Rewrite 2x^{2}-45x+232 as \left(2x^{2}-29x\right)+\left(-16x+232\right).

$$\left(2x^{2}-29x\right)+\left(-16x+232\right)$$

Factor out x in the first and -8 in the second group.

$$x\left(2x-29\right)-8\left(2x-29\right)$$

Factor out common term 2x-29 by using distributive property.

$$\left(2x-29\right)\left(x-8\right)$$

Rewrite the complete factored expression.

$$2\left(2x-29\right)\left(x-8\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$4x^{2}-90x+464=0$$

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.

$$x=\frac{-\left(-90\right)±\sqrt{\left(-90\right)^{2}-4\times 4\times 464}}{2\times 4}$$

Square -90.

$$x=\frac{-\left(-90\right)±\sqrt{8100-4\times 4\times 464}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-90\right)±\sqrt{8100-16\times 464}}{2\times 4}$$

Multiply -16 times 464.

$$x=\frac{-\left(-90\right)±\sqrt{8100-7424}}{2\times 4}$$

Add 8100 to -7424.

$$x=\frac{-\left(-90\right)±\sqrt{676}}{2\times 4}$$

Take the square root of 676.

$$x=\frac{-\left(-90\right)±26}{2\times 4}$$

The opposite of -90 is 90.

$$x=\frac{90±26}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{90±26}{8}$$

Now solve the equation x=\frac{90±26}{8} when ± is plus. Add 90 to 26.

$$x=\frac{116}{8}$$

Reduce the fraction \frac{116}{8} to lowest terms by extracting and canceling out 4.

$$x=\frac{29}{2}$$

Now solve the equation x=\frac{90±26}{8} when ± is minus. Subtract 26 from 90.

$$x=\frac{64}{8}$$

Divide 64 by 8.

$$x=8$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{29}{2} for x_{1} and 8 for x_{2}.

$$4x^{2}-90x+464=4\left(x-\frac{29}{2}\right)\left(x-8\right)$$

Subtract \frac{29}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$4x^{2}-90x+464=4\times \frac{2x-29}{2}\left(x-8\right)$$

Cancel out 2, the greatest common factor in 4 and 2.

$$4x^{2}-90x+464=2\left(2x-29\right)\left(x-8\right)$$

Evaluate

$$2\left(x-8\right)\left(2x-29\right)$$