Solve 4r^2+37r+63

$$4 r ^ { 2 } + 37 r + 63$$

Factor

$$\left(r+7\right)\left(4r+9\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 4r^{2}+ar+br+63. To find a and b, set up a system to be solved.

$$a+b=37 ab=4\times 63=252$$

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 252.

$$1,252 2,126 3,84 4,63 6,42 7,36 9,28 12,21 14,18$$

Calculate the sum for each pair.

$$1+252=253 2+126=128 3+84=87 4+63=67 6+42=48 7+36=43 9+28=37 12+21=33 14+18=32$$

The solution is the pair that gives sum 37.

$$a=9 b=28$$

Rewrite 4r^{2}+37r+63 as \left(4r^{2}+9r\right)+\left(28r+63\right).

$$\left(4r^{2}+9r\right)+\left(28r+63\right)$$

Factor out r in the first and 7 in the second group.

$$r\left(4r+9\right)+7\left(4r+9\right)$$

Factor out common term 4r+9 by using distributive property.

$$\left(4r+9\right)\left(r+7\right)$$

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.

$$4r^{2}+37r+63=0$$

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.

$$r=\frac{-37±\sqrt{37^{2}-4\times 4\times 63}}{2\times 4}$$

Square 37.

$$r=\frac{-37±\sqrt{1369-4\times 4\times 63}}{2\times 4}$$

Multiply -4 times 4.

$$r=\frac{-37±\sqrt{1369-16\times 63}}{2\times 4}$$

Multiply -16 times 63.

$$r=\frac{-37±\sqrt{1369-1008}}{2\times 4}$$

Add 1369 to -1008.

$$r=\frac{-37±\sqrt{361}}{2\times 4}$$

Take the square root of 361.

$$r=\frac{-37±19}{2\times 4}$$

Multiply 2 times 4.

$$r=\frac{-37±19}{8}$$

Now solve the equation r=\frac{-37±19}{8} when ± is plus. Add -37 to 19.

$$r=-\frac{18}{8}$$

Reduce the fraction \frac{-18}{8} to lowest terms by extracting and canceling out 2.

$$r=-\frac{9}{4}$$

Now solve the equation r=\frac{-37±19}{8} when ± is minus. Subtract 19 from -37.

$$r=-\frac{56}{8}$$

Divide -56 by 8.

$$r=-7$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{9}{4} for x_{1} and -7 for x_{2}.

$$4r^{2}+37r+63=4\left(r-\left(-\frac{9}{4}\right)\right)\left(r-\left(-7\right)\right)$$

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

$$4r^{2}+37r+63=4\left(r+\frac{9}{4}\right)\left(r+7\right)$$

Add \frac{9}{4} to r by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$4r^{2}+37r+63=4\times \frac{4r+9}{4}\left(r+7\right)$$

Cancel out 4, the greatest common factor in 4 and 4.

$$4r^{2}+37r+63=\left(4r+9\right)\left(r+7\right)$$

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 4

$$x ^ 2 +\frac{37}{4}x +\frac{63}{4} = 0$$

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

$$r + s = -\frac{37}{4} rs = \frac{63}{4}$$

Two numbers r and s sum up to -\frac{37}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{37}{4} = -\frac{37}{8}. 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>

$$r = -\frac{37}{8} - u s = -\frac{37}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{63}{4}

$$(-\frac{37}{8} - u) (-\frac{37}{8} + u) = \frac{63}{4}$$

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

$$\frac{1369}{64} - u^2 = \frac{63}{4}$$

Simplify the expression by subtracting \frac{1369}{64} on both sides

$$-u^2 = \frac{63}{4}-\frac{1369}{64} = -\frac{361}{64}$$

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

$$u^2 = \frac{361}{64} u = \pm\sqrt{\frac{361}{64}} = \pm \frac{19}{8}$$

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

$$r =-\frac{37}{8} - \frac{19}{8} = -7 s = -\frac{37}{8} + \frac{19}{8} = -2.250$$

Evaluate

$$\left(r+7\right)\left(4r+9\right)$$