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 terms with coefficients, without fully solving the equation. This approach streamlines problem-solving and deepens comprehension of polynomial behavior.

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 equation. The term under the square root, called the discriminant (\(\Delta = b^2 - 4ac\)), determines the nature and number of 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 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 case, for the quadratic \)x^2 - 8x + 16 = 0\(, where \)a = 1\(, \)b = -8\(, and \)c = 16\(, the discriminant is zero:

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

This means there is only one solution, \)x = 4$, which is also called a repeated or double root.

Understanding how to apply the quadratic formula and interpret the discriminant is essential for solving any quadratic equation, especially when factoring or completing the square becomes cumbersome. Always remember to substitute the coefficients carefully and simplify step-by-step. Additionally, verifying your solutions by plugging them back into the original equation ensures accuracy and reinforces comprehension of quadratic functions and their roots.

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. $z=\frac43,z=-\frac16$
• B. $z=\frac43,z=-\frac12$
• C. $z=\frac54,z=-\frac12$
• D. $z=\frac54,z=-\frac16$

Answer: (B)

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

• A. $x=\frac12$
• B. $x=-\frac12$
• C. $x=1,x=\frac12$
• D. $x=1,x=-\frac12$

Answer: (A)

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

• A. $x=5,x=\frac12$
• B. $x=5,x=\frac14$
• C. $x=4,x=\frac14$
• D. $x=4,x=\frac12$

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. $x=-\frac{13}{3},x=\frac12$
• 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 universal method to solve any quadratic equation of the form $a{x}^2+bx+c=0$. 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 your quadratic equation, substitute them into the formula, and simplify. This will give you the solutions for $x$. The plus or minus sign means there can be two solutions, one solution, or none, depending on the value inside the square root.

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 $D=b^2-4ac$. It tells us the nature and number of solutions of the quadratic equation. If $D>\; 0$, there are two distinct real solutions. If $D=\; 0$, there is exactly one real solution (a repeated root). If , there are no real solutions, but two imaginary (complex) solutions. This helps quickly determine the type of roots without fully solving the equation.

How do you simplify the quadratic formula when the discriminant is a perfect square?

When the discriminant $D=b^2-4ac$ is a perfect square, the square root simplifies to an integer or rational number. This makes the quadratic formula easier to simplify because you can directly add and subtract this value from $-b$ and then divide by $2a$. For example, if $D=\; 49$, then $\sqrt{49}=\; 7$. This leads to two rational solutions, which can be simplified further if possible. Perfect square discriminants often indicate the quadratic factors nicely.

Why might the quadratic formula be preferred over completing the square or factoring?

The quadratic formula is preferred because it works for all quadratic equations, regardless of the coefficients. Completing the square can become complicated if $a$ is not 1 or if $b$ is odd, and factoring only works easily when the quadratic factors nicely into integers. The quadratic formula provides a straightforward, reliable method to find solutions without guessing or trial and error, making it especially useful for complex or non-factorable quadratics.

How do you interpret the solutions when the quadratic formula gives one solution?

When the quadratic formula yields one solution, it means the discriminant is zero ($b^2-4ac=\; 0$). This indicates the quadratic has a repeated root, or a perfect square trinomial. The graph of the quadratic touches the x-axis at exactly one point, called the vertex. The single solution is $x=\; -\frac{b}{2a}$. This means the parabola is tangent to the x-axis.

What does it mean if the quadratic formula results in imaginary solutions?

If the quadratic formula results in imaginary solutions, it means the discriminant is negative (). Since you cannot take the square root of a negative number in the real number system, the solutions involve imaginary numbers. These solutions are complex conjugates and indicate the parabola does not intersect the x-axis. Imaginary solutions are written using the imaginary unit $i$, where $\sqrt{-1}=i$.