The Quadratic Formula: Videos & Practice Problems

The quadratic formula, $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, solves any quadratic equation in standard form by substituting coefficients a, b, and c. The discriminant, $b^2-4ac$, reveals the nature of solutions: positive indicates two real solutions, zero one real solution, and negative two imaginary solutions. Understanding these concepts aids in analyzing polynomials, including perfect square trinomials, and efficiently determining solution types without full equation solving.

Introduction to the Quadratic Formula

Solving quadratic equations can be efficiently done using the quadratic formula, which provides solutions for any quadratic equation in standard form \(ax^2 + bx + c = 0\). The quadratic formula is expressed as:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

This formula allows you to find the roots of the quadratic equation by substituting the coefficients \(a\), \(b\), and \(c\) directly into the formula. The term under the square root, \(b^2 - 4ac\), is called the discriminant and plays a crucial role in determining the nature of the solutions. If the discriminant is positive, there are two distinct real solutions; if it is zero, there is exactly one real solution; and if it is negative, the solutions are complex or imaginary.

For example, consider the quadratic equation \$2x^2 - 3x - 5 = 0\(. Here, \)a = 2\(, \)b = -3\(, and \)c = -5\(. Plugging these values into the quadratic formula gives:

\[x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-5)}}{2(2)} = \frac{3 \pm \sqrt{9 + 40}}{4} = \frac{3 \pm \sqrt{49}}{4} = \frac{3 \pm 7}{4}\]

This results in two solutions:

\[x = \frac{3 + 7}{4} = \frac{10}{4} = \frac{5}{2} \quad \text{and} \quad x = \frac{3 - 7}{4} = \frac{-4}{4} = -1\]

In another example, the quadratic equation \)x^2 - 8x + 16 = 0\( has \)a = 1\(, \)b = -8\(, and \)c = 16\(. Substituting these values yields:

\[x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(16)}}{2(1)} = \frac{8 \pm \sqrt{64 - 64}}{2} = \frac{8 \pm 0}{2} = \frac{8}{2} = 4\]

Here, the discriminant is zero, so there is only one real solution, \)x = 4$.

Understanding how to apply the quadratic formula and interpret the discriminant helps in solving any quadratic equation systematically. Always remember to verify your solutions by substituting them back into the original equation to ensure accuracy. Mastery of this formula is essential for solving quadratic equations efficiently, especially when factoring or completing the square becomes cumbersome.

Find the solution(s) using the quadratic formula.
$x^2+6x-7=0$

• A. $x=2,x=-7$
• B. $x=1,x=-3$
• C. $x=1,x=-7$
• D. $x=2,x=-3$

Answer: (C)

Find the solution(s) using the quadratic formula.
$\frac{3}{2}{z}^2-\frac{5}{4}z-1=0$

• A. $z=\frac{4}{3},z=-\frac{1}{6}$
• B. $z=\frac{4}{3},z=-\frac{1}{2}$
• C. $z=\frac{5}{4},z=-\frac{1}{2}$
• D. $z=\frac{5}{4},z=-\frac{1}{6}$

Answer: (B)

Find the solution(s) using the quadratic formula.
$4x^2-4x+1=0$

• A. $x=\frac{1}{2}$
• B. $x=-\frac{1}{2}$
• C. $x=1,x=\frac{1}{2}$
• D. $x=1,x=-\frac{1}{2}$

Answer: (A)

Find the solution(s) using the quadratic formula.
$4{(x-2)}^2-5=x+7$

• A. $x=5,x=\frac{1}{2}$
• B. $x=5,x=\frac{1}{4}$
• C. $x=4,x=\frac{1}{4}$
• D. $x=4,x=\frac{1}{2}$

Answer: (C)

Find the solution(s) using the quadratic formula.
$\frac{3}{2}{(x+5)}^2-4=x+1$

• A. $x=\frac{13}{x},x=-5$
• B. $x=-\frac{13}{3},x=-5$
• C. $x=-\frac{13}{3},x=\frac{1}{2}$
• D. $x=-5,x=5$

Answer: (B)

The Discriminant

The discriminant is a crucial concept in understanding the nature of solutions for quadratic equations. It is defined as the expression under the square root in the quadratic formula, represented mathematically as \(D = b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation in the standard form \(ax^2 + bx + c = 0\).

The value of the discriminant provides insight into the number and type of solutions the quadratic equation has:

• If \(D > 0\), the equation has two distinct real solutions.
• If \(D = 0\), there is exactly one real solution, often referred to as a repeated or double root.
• If \(D < 0\), the equation has no real solutions, but instead has two complex (imaginary) solutions.

To illustrate how to use the discriminant, consider the following examples:

1. For the quadratic equation \(2x^2 + 3x - 2 = 0\), we identify \(a = 2\), \(b = 3\), and \(c = -2\). Plugging these values into the discriminant formula gives:

\[D = 3^2 - 4 \cdot 2 \cdot (-2) = 9 + 16 = 25\]

Since \(D = 25\) (which is positive), this equation has two real solutions.

2. In the case of \(4x^2 + x + 2 = 0\), we have \(a = 4\), \(b = 1\), and \(c = 2\). The discriminant calculation is:

\[D = 1^2 - 4 \cdot 4 \cdot 2 = 1 - 32 = -31\]

Here, \(D = -31\) (negative), indicating that there are no real solutions, only two imaginary solutions.

3. Finally, for the equation \(x^2 - 10x + 25 = 0\), we find \(a = 1\), \(b = -10\), and \(c = 25\). The discriminant is calculated as follows:

\[D = (-10)^2 - 4 \cdot 1 \cdot 25 = 100 - 100 = 0\]

With \(D = 0\), this equation has one real solution.

Understanding the discriminant allows for quick assessments of quadratic equations without the need for complete solutions, making it a valuable tool in algebra.

Determine the number and type of solutions of the given quadratic equation. Do not solve. 
$x^2+8x+16=0$

• A. 2 real solutions
• B. 1 real solution
• C. 2 imaginary solutions

Answer: (B)

Determine the number and type of solutions of the given quadratic equation. Do not solve. 
$-4x^2+4x+5=0$

• A. 2 real solutions 
• B. 1 real solution 
• C. 2 imaginary solutions 

Answer: (A)

Here’s what students ask on this topic:

What is the quadratic formula and how do you use it to solve quadratic equations?

The quadratic formula is a method to find the solutions of any quadratic equation in standard form, which is $\mathrm{ax}{x}^2+\mathrm{bx}+c=0$. The formula is given by $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$. To use it, identify the coefficients $a$, $b$, and $c$ from the quadratic equation, then substitute them into the formula. Simplify the expression under the square root (called the discriminant), then calculate the two possible values for $x$ by adding and subtracting the square root term. This method works for all quadratic equations, even when factoring is difficult.

What is the discriminant in the quadratic formula and what does it tell us about the solutions?

The discriminant is the part of the quadratic formula under the square root, expressed as $b^2-4ac$. It tells us the number and type of solutions of the quadratic equation. If the discriminant is positive, there are two distinct real solutions. If it is zero, there is exactly one real solution (a repeated root). If the discriminant is negative, there are no real solutions, but two imaginary (complex) solutions. This helps quickly determine the nature of the roots without fully solving the equation.

How do you solve a quadratic equation using the quadratic formula when the discriminant is zero?

When the discriminant $b^2-4ac$ equals zero, the quadratic formula simplifies because the square root term becomes zero. This means the formula reduces to $x=\frac{-b}{2a}$. In this case, there is only one real solution, also called a repeated root. This happens when the quadratic is a perfect square trinomial, and the graph of the quadratic touches the x-axis at exactly one point.

Can the quadratic formula be used to solve any quadratic equation, even if the coefficients are not integers?

Yes, the quadratic formula can solve any quadratic equation in standard form, regardless of whether the coefficients $a$, $b$, and $c$ are integers, fractions, or decimals. This is one of its advantages over factoring or completing the square, which can be complicated with non-integer coefficients. By substituting the coefficients into the formula and simplifying carefully, you can find the exact or approximate roots of the quadratic equation.

How do you check if the solutions obtained from the quadratic formula are correct?

To verify solutions from the quadratic formula, substitute each solution back into the original quadratic equation. Replace $x$ with the solution value and simplify both sides. If the equation balances (both sides are equal), the solution is correct. This step is important because it confirms that the roots satisfy the equation, especially when dealing with complex or decimal answers.