Solve 45x^2-8x-21

$$45 x ^ { 2 } - 8 x - 21$$

Factor

$$\left(9x-7\right)\left(5x+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 45x^{2}+ax+bx-21. To find a and b, set up a system to be solved.

$$a+b=-8 ab=45\left(-21\right)=-945$$

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 -945.

$$1,-945 3,-315 5,-189 7,-135 9,-105 15,-63 21,-45 27,-35$$

Calculate the sum for each pair.

$$1-945=-944 3-315=-312 5-189=-184 7-135=-128 9-105=-96 15-63=-48 21-45=-24 27-35=-8$$

The solution is the pair that gives sum -8.

$$a=-35 b=27$$

Rewrite 45x^{2}-8x-21 as \left(45x^{2}-35x\right)+\left(27x-21\right).

$$\left(45x^{2}-35x\right)+\left(27x-21\right)$$

Factor out 5x in the first and 3 in the second group.

$$5x\left(9x-7\right)+3\left(9x-7\right)$$

Factor out common term 9x-7 by using distributive property.

$$\left(9x-7\right)\left(5x+3\right)$$

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.

$$45x^{2}-8x-21=0$$

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.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 45\left(-21\right)}}{2\times 45}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 45\left(-21\right)}}{2\times 45}$$

Multiply -4 times 45.

$$x=\frac{-\left(-8\right)±\sqrt{64-180\left(-21\right)}}{2\times 45}$$

Multiply -180 times -21.

$$x=\frac{-\left(-8\right)±\sqrt{64+3780}}{2\times 45}$$

Add 64 to 3780.

$$x=\frac{-\left(-8\right)±\sqrt{3844}}{2\times 45}$$

Take the square root of 3844.

$$x=\frac{-\left(-8\right)±62}{2\times 45}$$

The opposite of -8 is 8.

$$x=\frac{8±62}{2\times 45}$$

Multiply 2 times 45.

$$x=\frac{8±62}{90}$$

Now solve the equation x=\frac{8±62}{90} when ± is plus. Add 8 to 62.

$$x=\frac{70}{90}$$

Reduce the fraction \frac{70}{90} to lowest terms by extracting and canceling out 10.

$$x=\frac{7}{9}$$

Now solve the equation x=\frac{8±62}{90} when ± is minus. Subtract 62 from 8.

$$x=-\frac{54}{90}$$

Reduce the fraction \frac{-54}{90} to lowest terms by extracting and canceling out 18.

$$x=-\frac{3}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7}{9} for x_{1} and -\frac{3}{5} for x_{2}.

$$45x^{2}-8x-21=45\left(x-\frac{7}{9}\right)\left(x-\left(-\frac{3}{5}\right)\right)$$

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

$$45x^{2}-8x-21=45\left(x-\frac{7}{9}\right)\left(x+\frac{3}{5}\right)$$

Subtract \frac{7}{9} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$45x^{2}-8x-21=45\times \frac{9x-7}{9}\left(x+\frac{3}{5}\right)$$

Add \frac{3}{5} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$45x^{2}-8x-21=45\times \frac{9x-7}{9}\times \frac{5x+3}{5}$$

Multiply \frac{9x-7}{9} times \frac{5x+3}{5} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$45x^{2}-8x-21=45\times \frac{\left(9x-7\right)\left(5x+3\right)}{9\times 5}$$

Multiply 9 times 5.

$$45x^{2}-8x-21=45\times \frac{\left(9x-7\right)\left(5x+3\right)}{45}$$

Cancel out 45, the greatest common factor in 45 and 45.

$$45x^{2}-8x-21=\left(9x-7\right)\left(5x+3\right)$$

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 45

$$x ^ 2 -\frac{8}{45}x -\frac{7}{15} = 0$$

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

$$r + s = \frac{8}{45} rs = -\frac{7}{15}$$

Two numbers r and s sum up to \frac{8}{45} exactly when the average of the two numbers is \frac{1}{2}*\frac{8}{45} = \frac{4}{45}. 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>

$$r = \frac{4}{45} - u s = \frac{4}{45} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{7}{15}

$$(\frac{4}{45} - u) (\frac{4}{45} + u) = -\frac{7}{15}$$

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

$$\frac{16}{2025} - u^2 = -\frac{7}{15}$$

Simplify the expression by subtracting \frac{16}{2025} on both sides

$$-u^2 = -\frac{7}{15}-\frac{16}{2025} = -\frac{961}{2025}$$

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

$$u^2 = \frac{961}{2025} u = \pm\sqrt{\frac{961}{2025}} = \pm \frac{31}{45}$$

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

$$r =\frac{4}{45} - \frac{31}{45} = -0.600 s = \frac{4}{45} + \frac{31}{45} = 0.778$$

Evaluate

$$\left(9x-7\right)\left(5x+3\right)$$