Solve 6x^2+7x-49

$$6 x ^ { 2 } + 7 x - 49$$

Factor

$$\left(3x-7\right)\left(2x+7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 6x^{2}+ax+bx-49. To find a and b, set up a system to be solved.

$$a+b=7 ab=6\left(-49\right)=-294$$

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 -294.

$$-1,294 -2,147 -3,98 -6,49 -7,42 -14,21$$

Calculate the sum for each pair.

$$-1+294=293 -2+147=145 -3+98=95 -6+49=43 -7+42=35 -14+21=7$$

The solution is the pair that gives sum 7.

$$a=-14 b=21$$

Rewrite 6x^{2}+7x-49 as \left(6x^{2}-14x\right)+\left(21x-49\right).

$$\left(6x^{2}-14x\right)+\left(21x-49\right)$$

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

$$2x\left(3x-7\right)+7\left(3x-7\right)$$

Factor out common term 3x-7 by using distributive property.

$$\left(3x-7\right)\left(2x+7\right)$$

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.

$$6x^{2}+7x-49=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\times 6\left(-49\right)}}{2\times 6}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\times 6\left(-49\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-7±\sqrt{49-24\left(-49\right)}}{2\times 6}$$

Multiply -24 times -49.

$$x=\frac{-7±\sqrt{49+1176}}{2\times 6}$$

Add 49 to 1176.

$$x=\frac{-7±\sqrt{1225}}{2\times 6}$$

Take the square root of 1225.

$$x=\frac{-7±35}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-7±35}{12}$$

Now solve the equation x=\frac{-7±35}{12} when ± is plus. Add -7 to 35.

$$x=\frac{28}{12}$$

Reduce the fraction \frac{28}{12} to lowest terms by extracting and canceling out 4.

$$x=\frac{7}{3}$$

Now solve the equation x=\frac{-7±35}{12} when ± is minus. Subtract 35 from -7.

$$x=-\frac{42}{12}$$

Reduce the fraction \frac{-42}{12} to lowest terms by extracting and canceling out 6.

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

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

$$6x^{2}+7x-49=6\left(x-\frac{7}{3}\right)\left(x-\left(-\frac{7}{2}\right)\right)$$

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

$$6x^{2}+7x-49=6\left(x-\frac{7}{3}\right)\left(x+\frac{7}{2}\right)$$

Subtract \frac{7}{3} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$6x^{2}+7x-49=6\times \frac{3x-7}{3}\left(x+\frac{7}{2}\right)$$

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

$$6x^{2}+7x-49=6\times \frac{3x-7}{3}\times \frac{2x+7}{2}$$

Multiply \frac{3x-7}{3} times \frac{2x+7}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$6x^{2}+7x-49=6\times \frac{\left(3x-7\right)\left(2x+7\right)}{3\times 2}$$

Multiply 3 times 2.

$$6x^{2}+7x-49=6\times \frac{\left(3x-7\right)\left(2x+7\right)}{6}$$

Cancel out 6, the greatest common factor in 6 and 6.

$$6x^{2}+7x-49=\left(3x-7\right)\left(2x+7\right)$$

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 6

$$x ^ 2 +\frac{7}{6}x -\frac{49}{6} = 0$$

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

$$r + s = -\frac{7}{6} rs = -\frac{49}{6}$$

Two numbers r and s sum up to -\frac{7}{6} exactly when the average of the two numbers is \frac{1}{2}*-\frac{7}{6} = -\frac{7}{12}. 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>

$$r = -\frac{7}{12} - u s = -\frac{7}{12} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{49}{6}

$$(-\frac{7}{12} - u) (-\frac{7}{12} + u) = -\frac{49}{6}$$

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

$$\frac{49}{144} - u^2 = -\frac{49}{6}$$

Simplify the expression by subtracting \frac{49}{144} on both sides

$$-u^2 = -\frac{49}{6}-\frac{49}{144} = -\frac{1225}{144}$$

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

$$u^2 = \frac{1225}{144} u = \pm\sqrt{\frac{1225}{144}} = \pm \frac{35}{12}$$

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

$$r =-\frac{7}{12} - \frac{35}{12} = -3.500 s = -\frac{7}{12} + \frac{35}{12} = 2.333$$

Evaluate

$$\left(3x-7\right)\left(2x+7\right)$$