Equation Solver
29 Mar

Solving quadratic equations by completing the square method is important and understanding it helps a lot.

Completing the square x+a2=x2+2ax+a2 is the process of rearranging one side of a quadratic equation into a perfect square before solving and is achieved by following these simple rules: 



Problem 1

Solve this equation using completing the square

\huge 2x^2+5x+3=0{\color{Magenta} }

Step One

Simplify the expression where possible by dividing throughout by the coefficient of x2

\huge \frac{2}{2}x^2+\frac{5}{2}x+\frac{3}{2}=0
\huge x^2+\frac{5}{2}x+\frac{3}{2}=0

Step Two

Rearrange the new equation so that x2 and x terms are on left side of the equal sign (=) i.e LHS (Left Hand Side), and the constants on the right side of the equal sign i.e RHS (Right Hand Side)
\huge x^2+\frac{5}{2}x=\left(\frac{-3}{2}\right)

LHS = RHS

Step Three

Add to both sides of the equation, half the coefficient of x where coefficient of x = (2.5)
\huge x^2+\frac{5}{2}x+\left(\frac{5}{4}\right)^2=\frac{-3}{2}+\left(\frac{5}{4}\right)^2

Make a perfect square from the expression and then split the terms into LHS and RHS sides in order to solve for x.

\huge \left[x+\left(\frac{5}{4}\right)\right]^2=\frac{-3}{2}+\left(\frac{5}{4}\right)^2
LHS expression is on the left hand side and R.H.S expression on the right hand side

Evaluate the LHS and make it a perfect square like so:

\huge \left[x+\left(\frac{5}{4}\right)\right]^2 =\left(x+\frac{5}{4}\right)^2

Evaluate the RHS

\huge RHS\approx \frac{-3}{2}+\left(\frac{5}{4}\right)^2=\frac{8}{127}

Step Four

Now combine the two sides LHS and RHS respectively to resolve for value of x

\huge \left(x+\frac{5}{4}\right)^2=\frac{8}{127}
\huge x+\frac{5}{4}=\sqrt{\frac{8}{127}}
\huge x+\frac{5}{4}=\pm 0.251
\huge x+\frac{5}{4}-\frac{5}{4}=\pm 0.251-\frac{5}{4}
\huge x=-1.25\pm 0.25

The roots of the equations are

\huge {\color{DarkGreen} x_1=-1.25+0.25=-1}
\huge {\color{DarkGreen} x_2=-1.25- 0.25 = -1.5}

Basic Guidelines and Rules of Completing The Square

1. Begin with a quadratic equation in monic standard form  ax2+bx+c , otherwise, simplify the equation until the coefficient of x2 is reduced to unity.

2. Divide each side by a, the coefficient of the squared term.

3. Subtract the constant terms from both sides of the equation 

4. Rearrange the equations so that the x2 and x terms are on one side of the equals sign and the constant is on the other side.

5. Add the square of one-half of 

ca

, the coefficient of x, to both sides and make it the left side into a perfect square.

6. Evaluate the left side as square and simplify the right hand side where necessary.

7. Take the square root of both sides of the equation (remembering that the square root of a number gives a ± answer).

8. Solve the two linear equations with respect to ±