Solve 4a^2+9a+2

$$4 a ^ { 2 } + 9 a + 2$$

Factor

$$\left(a+2\right)\left(4a+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 4a^{2}+pa+qa+2. To find p and q, set up a system to be solved.

$$p+q=9 pq=4\times 2=8$$

Since pq is positive, p and q have the same sign. Since p+q is positive, p and q are both positive. List all such integer pairs that give product 8.

$$1,8 2,4$$

Calculate the sum for each pair.

$$1+8=9 2+4=6$$

The solution is the pair that gives sum 9.

$$p=1 q=8$$

Rewrite 4a^{2}+9a+2 as \left(4a^{2}+a\right)+\left(8a+2\right).

$$\left(4a^{2}+a\right)+\left(8a+2\right)$$

Factor out a in the first and 2 in the second group.

$$a\left(4a+1\right)+2\left(4a+1\right)$$

Factor out common term 4a+1 by using distributive property.

$$\left(4a+1\right)\left(a+2\right)$$

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.

$$4a^{2}+9a+2=0$$

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.

$$a=\frac{-9±\sqrt{9^{2}-4\times 4\times 2}}{2\times 4}$$

Square 9.

$$a=\frac{-9±\sqrt{81-4\times 4\times 2}}{2\times 4}$$

Multiply -4 times 4.

$$a=\frac{-9±\sqrt{81-16\times 2}}{2\times 4}$$

Multiply -16 times 2.

$$a=\frac{-9±\sqrt{81-32}}{2\times 4}$$

Add 81 to -32.

$$a=\frac{-9±\sqrt{49}}{2\times 4}$$

Take the square root of 49.

$$a=\frac{-9±7}{2\times 4}$$

Multiply 2 times 4.

$$a=\frac{-9±7}{8}$$

Now solve the equation a=\frac{-9±7}{8} when ± is plus. Add -9 to 7.

$$a=-\frac{2}{8}$$

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

$$a=-\frac{1}{4}$$

Now solve the equation a=\frac{-9±7}{8} when ± is minus. Subtract 7 from -9.

$$a=-\frac{16}{8}$$

Divide -16 by 8.

$$a=-2$$

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

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

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

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

Add \frac{1}{4} to a by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

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

$$4a^{2}+9a+2=\left(4a+1\right)\left(a+2\right)$$

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

$$x ^ 2 +\frac{9}{4}x +\frac{1}{2} = 0$$

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

$$r + s = -\frac{9}{4} rs = \frac{1}{2}$$

Two numbers r and s sum up to -\frac{9}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{9}{4} = -\frac{9}{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>

$$r = -\frac{9}{8} - u s = -\frac{9}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{1}{2}

$$(-\frac{9}{8} - u) (-\frac{9}{8} + u) = \frac{1}{2}$$

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

$$\frac{81}{64} - u^2 = \frac{1}{2}$$

Simplify the expression by subtracting \frac{81}{64} on both sides

$$-u^2 = \frac{1}{2}-\frac{81}{64} = -\frac{49}{64}$$

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

$$u^2 = \frac{49}{64} u = \pm\sqrt{\frac{49}{64}} = \pm \frac{7}{8}$$

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

$$r =-\frac{9}{8} - \frac{7}{8} = -2 s = -\frac{9}{8} + \frac{7}{8} = -0.250$$

Evaluate

$$\left(a+2\right)\left(4a+1\right)$$