Solve 56x^2-6x-17

$$56 { x }^{ 2 } -6x-17$$

Factor

$$\left(28x-17\right)\left(2x+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 56x^{2}+ax+bx-17. To find a and b, set up a system to be solved.

$$a+b=-6 ab=56\left(-17\right)=-952$$

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 -952.

$$1,-952 2,-476 4,-238 7,-136 8,-119 14,-68 17,-56 28,-34$$

Calculate the sum for each pair.

$$1-952=-951 2-476=-474 4-238=-234 7-136=-129 8-119=-111 14-68=-54 17-56=-39 28-34=-6$$

The solution is the pair that gives sum -6.

$$a=-34 b=28$$

Rewrite 56x^{2}-6x-17 as \left(56x^{2}-34x\right)+\left(28x-17\right).

$$\left(56x^{2}-34x\right)+\left(28x-17\right)$$

Factor out 2x in 56x^{2}-34x.

$$2x\left(28x-17\right)+28x-17$$

Factor out common term 28x-17 by using distributive property.

$$\left(28x-17\right)\left(2x+1\right)$$

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.

$$56x^{2}-6x-17=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 56\left(-17\right)}}{2\times 56}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 56\left(-17\right)}}{2\times 56}$$

Multiply -4 times 56.

$$x=\frac{-\left(-6\right)±\sqrt{36-224\left(-17\right)}}{2\times 56}$$

Multiply -224 times -17.

$$x=\frac{-\left(-6\right)±\sqrt{36+3808}}{2\times 56}$$

Add 36 to 3808.

$$x=\frac{-\left(-6\right)±\sqrt{3844}}{2\times 56}$$

Take the square root of 3844.

$$x=\frac{-\left(-6\right)±62}{2\times 56}$$

The opposite of -6 is 6.

$$x=\frac{6±62}{2\times 56}$$

Multiply 2 times 56.

$$x=\frac{6±62}{112}$$

Now solve the equation x=\frac{6±62}{112} when ± is plus. Add 6 to 62.

$$x=\frac{68}{112}$$

Reduce the fraction \frac{68}{112} to lowest terms by extracting and canceling out 4.

$$x=\frac{17}{28}$$

Now solve the equation x=\frac{6±62}{112} when ± is minus. Subtract 62 from 6.

$$x=-\frac{56}{112}$$

Reduce the fraction \frac{-56}{112} to lowest terms by extracting and canceling out 56.

$$x=-\frac{1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{17}{28} for x_{1} and -\frac{1}{2} for x_{2}.

$$56x^{2}-6x-17=56\left(x-\frac{17}{28}\right)\left(x-\left(-\frac{1}{2}\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$56x^{2}-6x-17=56\left(x-\frac{17}{28}\right)\left(x+\frac{1}{2}\right)$$

Subtract \frac{17}{28} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$56x^{2}-6x-17=56\times \frac{28x-17}{28}\left(x+\frac{1}{2}\right)$$

Add \frac{1}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$56x^{2}-6x-17=56\times \frac{28x-17}{28}\times \frac{2x+1}{2}$$

Multiply \frac{28x-17}{28} times \frac{2x+1}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$56x^{2}-6x-17=56\times \frac{\left(28x-17\right)\left(2x+1\right)}{28\times 2}$$

Multiply 28 times 2.

$$56x^{2}-6x-17=56\times \frac{\left(28x-17\right)\left(2x+1\right)}{56}$$

Cancel out 56, the greatest common factor in 56 and 56.

$$56x^{2}-6x-17=\left(28x-17\right)\left(2x+1\right)$$

Evaluate

$$\left(28x-17\right)\left(2x+1\right)$$