Solve 42x^2+372x-54

$$42 x ^ { 2 } + 372 x - 54$$

Factor

$$6\left(7x-1\right)\left(x+9\right)$$

Solution Steps

Factor out 6.

$$6\left(7x^{2}+62x-9\right)$$

Consider 7x^{2}+62x-9. Factor the expression by grouping. First, the expression needs to be rewritten as 7x^{2}+ax+bx-9. To find a and b, set up a system to be solved.

$$a+b=62 ab=7\left(-9\right)=-63$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -63.

$$-1,63 -3,21 -7,9$$

Calculate the sum for each pair.

$$-1+63=62 -3+21=18 -7+9=2$$

The solution is the pair that gives sum 62.

$$a=-1 b=63$$

Rewrite 7x^{2}+62x-9 as \left(7x^{2}-x\right)+\left(63x-9\right).

$$\left(7x^{2}-x\right)+\left(63x-9\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(7x-1\right)+9\left(7x-1\right)$$

Factor out common term 7x-1 by using distributive property.

$$\left(7x-1\right)\left(x+9\right)$$

Rewrite the complete factored expression.

$$6\left(7x-1\right)\left(x+9\right)$$

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.

$$42x^{2}+372x-54=0$$

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.

$$x=\frac{-372±\sqrt{372^{2}-4\times 42\left(-54\right)}}{2\times 42}$$

Square 372.

$$x=\frac{-372±\sqrt{138384-4\times 42\left(-54\right)}}{2\times 42}$$

Multiply -4 times 42.

$$x=\frac{-372±\sqrt{138384-168\left(-54\right)}}{2\times 42}$$

Multiply -168 times -54.

$$x=\frac{-372±\sqrt{138384+9072}}{2\times 42}$$

Add 138384 to 9072.

$$x=\frac{-372±\sqrt{147456}}{2\times 42}$$

Take the square root of 147456.

$$x=\frac{-372±384}{2\times 42}$$

Multiply 2 times 42.

$$x=\frac{-372±384}{84}$$

Now solve the equation x=\frac{-372±384}{84} when ± is plus. Add -372 to 384.

$$x=\frac{12}{84}$$

Reduce the fraction \frac{12}{84} to lowest terms by extracting and canceling out 12.

$$x=\frac{1}{7}$$

Now solve the equation x=\frac{-372±384}{84} when ± is minus. Subtract 384 from -372.

$$x=-\frac{756}{84}$$

Divide -756 by 84.

$$x=-9$$

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

$$42x^{2}+372x-54=42\left(x-\frac{1}{7}\right)\left(x-\left(-9\right)\right)$$

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

$$42x^{2}+372x-54=42\left(x-\frac{1}{7}\right)\left(x+9\right)$$

Subtract \frac{1}{7} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$42x^{2}+372x-54=42\times \frac{7x-1}{7}\left(x+9\right)$$

Cancel out 7, the greatest common factor in 42 and 7.

$$42x^{2}+372x-54=6\left(7x-1\right)\left(x+9\right)$$

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 42

$$x ^ 2 +\frac{62}{7}x -\frac{9}{7} = 0$$

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

$$r + s = -\frac{62}{7} rs = -\frac{9}{7}$$

Two numbers r and s sum up to -\frac{62}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{62}{7} = -\frac{31}{7}. 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>

$$r = -\frac{31}{7} - u s = -\frac{31}{7} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{9}{7}

$$(-\frac{31}{7} - u) (-\frac{31}{7} + u) = -\frac{9}{7}$$

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

$$\frac{961}{49} - u^2 = -\frac{9}{7}$$

Simplify the expression by subtracting \frac{961}{49} on both sides

$$-u^2 = -\frac{9}{7}-\frac{961}{49} = -\frac{1024}{49}$$

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

$$u^2 = \frac{1024}{49} u = \pm\sqrt{\frac{1024}{49}} = \pm \frac{32}{7}$$

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

$$r =-\frac{31}{7} - \frac{32}{7} = -9 s = -\frac{31}{7} + \frac{32}{7} = 0.143$$

Evaluate

$$6\left(7x-1\right)\left(x+9\right)$$