# Article: How to Solve Quadratic Equations
Quadratic equations are polynomial equations of the second degree. A general form of a quadratic equation is $ax^2+bx+c=0$, where a, b, and c are coefficients and ‘a’ should not be equal to zero.
Quadratic equations can be solved using different methods, which include
1. Factoring
2. The Quadratic Formula
3. Completing the Square
4. Graphically
## Factoring
The first method is factoring, which works best when the equation can easily be factored by hand. To solve a quadratic equation by factoring, we set the equation to zero, and then we rewrite the middle term as the sum of two factors that multiply to get ‘a*c’ and then add to ‘b’. Then, we factor by grouping and solve for the roots.
Example:
Solve $x^2+x-2=0$
Factoring to $(x+2)(x-1) = 0$
So, the roots are $x = -2$ and $x = 1$
## Quadratic Formula
The quadratic formula is applicable to any quadratic equation and is found when we complete the square on the general quadratic equation. The formula is expressed as $x=\frac{-b±\sqrt{b^2 – 4ac}}{2a}$. Here the symbol $\pm$ means that there are commonly two solutions for ‘x’, corresponding to the “+” and the “−” roots.
Example:
Solve $x^2+2x-15=0$
Applying the quadratic formula, you will get
$x=\frac{-b±\sqrt{b^2 – 4ac}}{2a}=\frac{-2±\sqrt{(2)^2 – 4(1)(-15)}}{2*1}=\frac{-2±\sqrt{4+60}}{2}=\frac{-2±\sqrt{64}}{2}=\frac{-2±8}{2}$
So, the roots are $x = -5$ and $x = 3$.
## Completing the Square
Completing the square is another method to solve quadratic equations. This process manipulates the quadratic equation into the form $(x-h)^2=k$.
Example:
Solve $x^2+12x+36=0$
Rearranging gives $(x+6)^2=0$
So, the root is $x = -6$
## Graphically
Graphically, the solutions to the quadratic equation are the x-intercepts of the parabola. Quadratic equations graph as curvy lines called parabolas that open either upward or downward.
# Questions About How to Solve Quadratic Equations
1. What are the four methods to solve quadratic equations?
2. What is the quadratic formula?
3. How to solve the quadratic equation by factoring?
4. What is the meaning of the ‘+’ and the ‘−’ in the quadratic formula?
5. What is the general form of a quadratic equation?
6. Why do we use completing the square method?
7. What is the ‘a’ in a quadratic equation?
8. What are the x-intercepts in the graphical method of solving quadratic equations?
9. Is it possible to solve all types of quadratic equations by factoring?
10. How can you solve $x^2 – 3x – 4 = 0$ by factoring?
11. Apply the quadratic formula to solve $3x^2 + 7x + 2 = 0$.
12. How to solve $x^2 + 4x + 4 = 0$ by completing the square?
13. How can you graph $x^2 – 6x + 8 = 0$?
14. Explain the process of factoring in solving quadratic equations.
15. How to choose the best method for solving a quadratic equation?
16. Can every quadratic equation be factored?
17. What is the importance of the ‘a’ coefficient in a quadratic equation?
18. How does the sign of ‘a’ affect the graph of a quadratic equation?
19. What is the discriminant in the quadratic formula?
20. Why should ‘a’ in the quadratic equation not be equal to zero?
# Answers for the Above Questions
1. The four methods to solve quadratic equations are factoring, using the quadratic formula, completing the square, and graphing.
2. The quadratic formula is $x=\frac{-b±\sqrt{b^2 – 4ac}}{2a}$.
3. To solve a quadratic equation by factoring, first set the equation to zero, then rewrite the middle term as the sum of two terms that multiply to get ‘a*c’ and then add to ‘b’.
4. The ‘+’ and the ‘−’ in the quadratic formula represent the two possible solutions for ‘x’.
5. The general form of a quadratic equation is $ax^2+bx+c=0$.
6. We use completing the square method when it is easier to manipulate the quadratic equation into the form $(x-h)^2=k$.
7. ‘a’ represents the coefficient of the squared term in a quadratic equation.
8. The x-intercepts in the graphical method of solving quadratic equations are the values of x where the graph crosses the x-axis, i.e., where the equation equals zero.
9. No, not all types of quadratic equations can be easily factored by hand.
10. To solve $x^2 – 3x – 4 = 0$ by factoring, the factors of -4 that add up to -3 are -4 and 1. So $x^2 – 3x – 4 = (x-4)(x+1)$, thus x =4 or x = -1.
11. Applying the quadratic formula to $3x^2 + 7x + 2 = 0$, we find $x=\frac{-7±\sqrt{(7)^2 – 4*3*2}}{2*3}=\frac{-7±\sqrt{49-24}}{6}=\frac{-7±\sqrt{25}}{6}=\frac{-7±5}{6}$ so the roots are $x=2$ and $x=-1/3$.
12. To solve $x^2 + 4x + 4 = 0$ by completing the square, rearrange to $(x+2)^2=0$. Thus, the root is $x=-2$.
13. To graph $x^2 – 6x + 8 = 0$, find the roots of the equation (x = 2 and x = 4) and plot them as the x-intercepts. The vertex of the parabola is at x = 3.
14. Factoring involves setting the equation to zero, and then rewriting the middle term as the sum of two factors that multiply to get ‘a*c’ and then add to ‘b’. Then factor by grouping and solve for the roots.
15. The choice depends on the given equation — simple ones can be factored easily, while more complex ones may require the quadratic formula.
16. No, not every quadratic equation can be factored, especially if the roots are not rational.
17. The ‘a’ coefficient in a quadratic equation determines whether the parabola opens upward (a > 0) or downward (a < 0).
18. If 'a' is positive, the graph of a quadratic equation opens upward. If 'a' is negative, it opens downward.
19. The discriminant in the quadratic formula is the part $b^2 - 4ac$. It determines the nature of the roots.
20. 'a' should not be equal to zero in the quadratic equation because if 'a' is zero, then the equation is linear, not quadratic.
