Solve 49x^2+63x+18

$$49 x ^ { 2 } + 63 x + 18$$

Factor

$$\left(7x+3\right)\left(7x+6\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 49x^{2}+ax+bx+18. To find a and b, set up a system to be solved.

$$a+b=63 ab=49\times 18=882$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 882.

$$1,882 2,441 3,294 6,147 7,126 9,98 14,63 18,49 21,42$$

Calculate the sum for each pair.

$$1+882=883 2+441=443 3+294=297 6+147=153 7+126=133 9+98=107 14+63=77 18+49=67 21+42=63$$

The solution is the pair that gives sum 63.

$$a=21 b=42$$

Rewrite 49x^{2}+63x+18 as \left(49x^{2}+21x\right)+\left(42x+18\right).

$$\left(49x^{2}+21x\right)+\left(42x+18\right)$$

Factor out 7x in the first and 6 in the second group.

$$7x\left(7x+3\right)+6\left(7x+3\right)$$

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

$$\left(7x+3\right)\left(7x+6\right)$$

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.

$$49x^{2}+63x+18=0$$

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.

$$x=\frac{-63±\sqrt{63^{2}-4\times 49\times 18}}{2\times 49}$$

Square 63.

$$x=\frac{-63±\sqrt{3969-4\times 49\times 18}}{2\times 49}$$

Multiply -4 times 49.

$$x=\frac{-63±\sqrt{3969-196\times 18}}{2\times 49}$$

Multiply -196 times 18.

$$x=\frac{-63±\sqrt{3969-3528}}{2\times 49}$$

Add 3969 to -3528.

$$x=\frac{-63±\sqrt{441}}{2\times 49}$$

Take the square root of 441.

$$x=\frac{-63±21}{2\times 49}$$

Multiply 2 times 49.

$$x=\frac{-63±21}{98}$$

Now solve the equation x=\frac{-63±21}{98} when ± is plus. Add -63 to 21.

$$x=-\frac{42}{98}$$

Reduce the fraction \frac{-42}{98} to lowest terms by extracting and canceling out 14.

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

Now solve the equation x=\frac{-63±21}{98} when ± is minus. Subtract 21 from -63.

$$x=-\frac{84}{98}$$

Reduce the fraction \frac{-84}{98} to lowest terms by extracting and canceling out 14.

$$x=-\frac{6}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{3}{7} for x_{1} and -\frac{6}{7} for x_{2}.

$$49x^{2}+63x+18=49\left(x-\left(-\frac{3}{7}\right)\right)\left(x-\left(-\frac{6}{7}\right)\right)$$

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

$$49x^{2}+63x+18=49\left(x+\frac{3}{7}\right)\left(x+\frac{6}{7}\right)$$

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

$$49x^{2}+63x+18=49\times \frac{7x+3}{7}\left(x+\frac{6}{7}\right)$$

Add \frac{6}{7} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$49x^{2}+63x+18=49\times \frac{7x+3}{7}\times \frac{7x+6}{7}$$

Multiply \frac{7x+3}{7} times \frac{7x+6}{7} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$49x^{2}+63x+18=49\times \frac{\left(7x+3\right)\left(7x+6\right)}{7\times 7}$$

Multiply 7 times 7.

$$49x^{2}+63x+18=49\times \frac{\left(7x+3\right)\left(7x+6\right)}{49}$$

Cancel out 49, the greatest common factor in 49 and 49.

$$49x^{2}+63x+18=\left(7x+3\right)\left(7x+6\right)$$

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 49

$$x ^ 2 +\frac{9}{7}x +\frac{18}{49} = 0$$

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

$$r + s = -\frac{9}{7} rs = \frac{18}{49}$$

Two numbers r and s sum up to -\frac{9}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{9}{7} = -\frac{9}{14}. 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>

$$r = -\frac{9}{14} - u s = -\frac{9}{14} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{18}{49}

$$(-\frac{9}{14} - u) (-\frac{9}{14} + u) = \frac{18}{49}$$

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

$$\frac{81}{196} - u^2 = \frac{18}{49}$$

Simplify the expression by subtracting \frac{81}{196} on both sides

$$-u^2 = \frac{18}{49}-\frac{81}{196} = -\frac{9}{196}$$

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

$$u^2 = \frac{9}{196} u = \pm\sqrt{\frac{9}{196}} = \pm \frac{3}{14}$$

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

$$r =-\frac{9}{14} - \frac{3}{14} = -0.857 s = -\frac{9}{14} + \frac{3}{14} = -0.429$$

Evaluate

$$\left(7x+3\right)\left(7x+6\right)$$