Solve 8x^2-53x-21=

$$8 x ^ { 2 } - 53 x - 21 =$$

Factor

$$\left(x-7\right)\left(8x+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 8x^{2}+ax+bx-21. To find a and b, set up a system to be solved.

$$a+b=-53 ab=8\left(-21\right)=-168$$

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 -168.

$$1,-168 2,-84 3,-56 4,-42 6,-28 7,-24 8,-21 12,-14$$

Calculate the sum for each pair.

$$1-168=-167 2-84=-82 3-56=-53 4-42=-38 6-28=-22 7-24=-17 8-21=-13 12-14=-2$$

The solution is the pair that gives sum -53.

$$a=-56 b=3$$

Rewrite 8x^{2}-53x-21 as \left(8x^{2}-56x\right)+\left(3x-21\right).

$$\left(8x^{2}-56x\right)+\left(3x-21\right)$$

Factor out 8x in the first and 3 in the second group.

$$8x\left(x-7\right)+3\left(x-7\right)$$

Factor out common term x-7 by using distributive property.

$$\left(x-7\right)\left(8x+3\right)$$

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.

$$8x^{2}-53x-21=0$$

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.

$$x=\frac{-\left(-53\right)±\sqrt{\left(-53\right)^{2}-4\times 8\left(-21\right)}}{2\times 8}$$

Square -53.

$$x=\frac{-\left(-53\right)±\sqrt{2809-4\times 8\left(-21\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-53\right)±\sqrt{2809-32\left(-21\right)}}{2\times 8}$$

Multiply -32 times -21.

$$x=\frac{-\left(-53\right)±\sqrt{2809+672}}{2\times 8}$$

Add 2809 to 672.

$$x=\frac{-\left(-53\right)±\sqrt{3481}}{2\times 8}$$

Take the square root of 3481.

$$x=\frac{-\left(-53\right)±59}{2\times 8}$$

The opposite of -53 is 53.

$$x=\frac{53±59}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{53±59}{16}$$

Now solve the equation x=\frac{53±59}{16} when ± is plus. Add 53 to 59.

$$x=\frac{112}{16}$$

Divide 112 by 16.

$$x=7$$

Now solve the equation x=\frac{53±59}{16} when ± is minus. Subtract 59 from 53.

$$x=-\frac{6}{16}$$

Reduce the fraction \frac{-6}{16} to lowest terms by extracting and canceling out 2.

$$x=-\frac{3}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 7 for x_{1} and -\frac{3}{8} for x_{2}.

$$8x^{2}-53x-21=8\left(x-7\right)\left(x-\left(-\frac{3}{8}\right)\right)$$

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

$$8x^{2}-53x-21=8\left(x-7\right)\left(x+\frac{3}{8}\right)$$

Add \frac{3}{8} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$8x^{2}-53x-21=8\left(x-7\right)\times \frac{8x+3}{8}$$

Cancel out 8, the greatest common factor in 8 and 8.

$$8x^{2}-53x-21=\left(x-7\right)\left(8x+3\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 8

$$x ^ 2 -\frac{53}{8}x -\frac{21}{8} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = \frac{53}{8} rs = -\frac{21}{8}$$

Two numbers r and s sum up to \frac{53}{8} exactly when the average of the two numbers is \frac{1}{2}*\frac{53}{8} = \frac{53}{16}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{53}{16} - u s = \frac{53}{16} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{21}{8}

$$(\frac{53}{16} - u) (\frac{53}{16} + u) = -\frac{21}{8}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{2809}{256} - u^2 = -\frac{21}{8}$$

Simplify the expression by subtracting \frac{2809}{256} on both sides

$$-u^2 = -\frac{21}{8}-\frac{2809}{256} = -\frac{3481}{256}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{3481}{256} u = \pm\sqrt{\frac{3481}{256}} = \pm \frac{59}{16}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =\frac{53}{16} - \frac{59}{16} = -0.375 s = \frac{53}{16} + \frac{59}{16} = 7$$

Evaluate

$$\left(x-7\right)\left(8x+3\right)$$