Solve 5r^2+4r-9

$$5 r ^ { 2 } + 4 r - 9$$

Factor

$$\left(r-1\right)\left(5r+9\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 5r^{2}+ar+br-9. To find a and b, set up a system to be solved.

$$a+b=4 ab=5\left(-9\right)=-45$$

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 -45.

$$-1,45 -3,15 -5,9$$

Calculate the sum for each pair.

$$-1+45=44 -3+15=12 -5+9=4$$

The solution is the pair that gives sum 4.

$$a=-5 b=9$$

Rewrite 5r^{2}+4r-9 as \left(5r^{2}-5r\right)+\left(9r-9\right).

$$\left(5r^{2}-5r\right)+\left(9r-9\right)$$

Factor out 5r in the first and 9 in the second group.

$$5r\left(r-1\right)+9\left(r-1\right)$$

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

$$\left(r-1\right)\left(5r+9\right)$$

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.

$$5r^{2}+4r-9=0$$

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.

$$r=\frac{-4±\sqrt{4^{2}-4\times 5\left(-9\right)}}{2\times 5}$$

Square 4.

$$r=\frac{-4±\sqrt{16-4\times 5\left(-9\right)}}{2\times 5}$$

Multiply -4 times 5.

$$r=\frac{-4±\sqrt{16-20\left(-9\right)}}{2\times 5}$$

Multiply -20 times -9.

$$r=\frac{-4±\sqrt{16+180}}{2\times 5}$$

Add 16 to 180.

$$r=\frac{-4±\sqrt{196}}{2\times 5}$$

Take the square root of 196.

$$r=\frac{-4±14}{2\times 5}$$

Multiply 2 times 5.

$$r=\frac{-4±14}{10}$$

Now solve the equation r=\frac{-4±14}{10} when ± is plus. Add -4 to 14.

$$r=\frac{10}{10}$$

Divide 10 by 10.

$$r=1$$

Now solve the equation r=\frac{-4±14}{10} when ± is minus. Subtract 14 from -4.

$$r=-\frac{18}{10}$$

Reduce the fraction \frac{-18}{10} to lowest terms by extracting and canceling out 2.

$$r=-\frac{9}{5}$$

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{9}{5} for x_{2}.

$$5r^{2}+4r-9=5\left(r-1\right)\left(r-\left(-\frac{9}{5}\right)\right)$$

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

$$5r^{2}+4r-9=5\left(r-1\right)\left(r+\frac{9}{5}\right)$$

Add \frac{9}{5} to r by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$5r^{2}+4r-9=5\left(r-1\right)\times \frac{5r+9}{5}$$

Cancel out 5, the greatest common factor in 5 and 5.

$$5r^{2}+4r-9=\left(r-1\right)\left(5r+9\right)$$

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 5

$$x ^ 2 +\frac{4}{5}x -\frac{9}{5} = 0$$

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

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

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

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

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{9}{5}

$$(-\frac{2}{5} - u) (-\frac{2}{5} + u) = -\frac{9}{5}$$

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

$$\frac{4}{25} - u^2 = -\frac{9}{5}$$

Simplify the expression by subtracting \frac{4}{25} on both sides

$$-u^2 = -\frac{9}{5}-\frac{4}{25} = -\frac{49}{25}$$

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

$$u^2 = \frac{49}{25} u = \pm\sqrt{\frac{49}{25}} = \pm \frac{7}{5}$$

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

$$r =-\frac{2}{5} - \frac{7}{5} = -1.800 s = -\frac{2}{5} + \frac{7}{5} = 1.000$$

Evaluate

$$\left(r-1\right)\left(5r+9\right)$$