Solve 12h^2+30h-42

$$12 h ^ { 2 } + 30 h - 42$$

Factor

$$6\left(h-1\right)\left(2h+7\right)$$

Solution Steps

Factor out 6.

$$6\left(2h^{2}+5h-7\right)$$

Consider 2h^{2}+5h-7. Factor the expression by grouping. First, the expression needs to be rewritten as 2h^{2}+ah+bh-7. To find a and b, set up a system to be solved.

$$a+b=5 ab=2\left(-7\right)=-14$$

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 -14.

$$-1,14 -2,7$$

Calculate the sum for each pair.

$$-1+14=13 -2+7=5$$

The solution is the pair that gives sum 5.

$$a=-2 b=7$$

Rewrite 2h^{2}+5h-7 as \left(2h^{2}-2h\right)+\left(7h-7\right).

$$\left(2h^{2}-2h\right)+\left(7h-7\right)$$

Factor out 2h in the first and 7 in the second group.

$$2h\left(h-1\right)+7\left(h-1\right)$$

Factor out common term h-1 by using distributive property.

$$\left(h-1\right)\left(2h+7\right)$$

Rewrite the complete factored expression.

$$6\left(h-1\right)\left(2h+7\right)$$

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.

$$12h^{2}+30h-42=0$$

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.

$$h=\frac{-30±\sqrt{30^{2}-4\times 12\left(-42\right)}}{2\times 12}$$

Square 30.

$$h=\frac{-30±\sqrt{900-4\times 12\left(-42\right)}}{2\times 12}$$

Multiply -4 times 12.

$$h=\frac{-30±\sqrt{900-48\left(-42\right)}}{2\times 12}$$

Multiply -48 times -42.

$$h=\frac{-30±\sqrt{900+2016}}{2\times 12}$$

Add 900 to 2016.

$$h=\frac{-30±\sqrt{2916}}{2\times 12}$$

Take the square root of 2916.

$$h=\frac{-30±54}{2\times 12}$$

Multiply 2 times 12.

$$h=\frac{-30±54}{24}$$

Now solve the equation h=\frac{-30±54}{24} when ± is plus. Add -30 to 54.

$$h=\frac{24}{24}$$

Divide 24 by 24.

$$h=1$$

Now solve the equation h=\frac{-30±54}{24} when ± is minus. Subtract 54 from -30.

$$h=-\frac{84}{24}$$

Reduce the fraction \frac{-84}{24} to lowest terms by extracting and canceling out 12.

$$h=-\frac{7}{2}$$

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

$$12h^{2}+30h-42=12\left(h-1\right)\left(h-\left(-\frac{7}{2}\right)\right)$$

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

$$12h^{2}+30h-42=12\left(h-1\right)\left(h+\frac{7}{2}\right)$$

Add \frac{7}{2} to h by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$12h^{2}+30h-42=12\left(h-1\right)\times \frac{2h+7}{2}$$

Cancel out 2, the greatest common factor in 12 and 2.

$$12h^{2}+30h-42=6\left(h-1\right)\left(2h+7\right)$$

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 12

$$x ^ 2 +\frac{5}{2}x -\frac{7}{2} = 0$$

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

$$r + s = -\frac{5}{2} rs = -\frac{7}{2}$$

Two numbers r and s sum up to -\frac{5}{2} exactly when the average of the two numbers is \frac{1}{2}*-\frac{5}{2} = -\frac{5}{4}. 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>

$$r = -\frac{5}{4} - u s = -\frac{5}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{7}{2}

$$(-\frac{5}{4} - u) (-\frac{5}{4} + u) = -\frac{7}{2}$$

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

$$\frac{25}{16} - u^2 = -\frac{7}{2}$$

Simplify the expression by subtracting \frac{25}{16} on both sides

$$-u^2 = -\frac{7}{2}-\frac{25}{16} = -\frac{81}{16}$$

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

$$u^2 = \frac{81}{16} u = \pm\sqrt{\frac{81}{16}} = \pm \frac{9}{4}$$

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

$$r =-\frac{5}{4} - \frac{9}{4} = -3.500 s = -\frac{5}{4} + \frac{9}{4} = 1$$

Evaluate

$$6\left(h-1\right)\left(2h+7\right)$$