Quadratic Equations & Applications: Videos & Practice Problems

The quadratic equation, expressed in $a{x}^2+bx+c=0$ standard form, models various real-world scenarios. Solving involves factoring the trinomial, setting each binomial factor equal to zero, and applying the zero product property. For example, factoring $x^2+10x+21=0$ yields $(x+3)(x+7)=0$, leading to solutions $x=-3$ and $x=-7$. Understanding coefficients, terms, and factoring techniques is essential for mastering polynomial equations.

Solving Quadratic Equations by Factoring

The quadratic equation is a fundamental mathematical tool used to model various real-world scenarios, such as the trajectory of projectiles and analyzing business sales versus costs. It is expressed in the standard form as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable to solve for. Solving a quadratic equation involves finding the values of \(x\) that satisfy this equation, making it equal to zero.

To solve a quadratic equation, the first step is to ensure it is written in standard form. Next, the equation is factored into two binomials. Factoring involves identifying two numbers that multiply to the constant term \(c\) and add up to the coefficient \(b\). For example, in the quadratic \(x^2 + 10x + 21 = 0\), the numbers 3 and 7 multiply to 21 and add to 10, allowing the equation to be factored as \((x + 3)(x + 7) = 0\).

Once factored, the zero product property is applied. This property states that if the product of two factors equals zero, then at least one of the factors must be zero. Therefore, setting each factor equal to zero gives the equations \(x + 3 = 0\) and \(x + 7 = 0\). Solving these yields the solutions \(x = -3\) and \(x = -7\).

Verification of these solutions involves substituting them back into the original quadratic equation to confirm that they satisfy it. For \(x = -3\), substituting yields \((-3)^2 + 10(-3) + 21 = 9 - 30 + 21 = 0\), confirming it as a valid solution. Similarly, for \(x = -7\), substitution results in \((-7)^2 + 10(-7) + 21 = 49 - 70 + 21 = 0\), also confirming its validity.

Mastering the process of writing quadratics in standard form, factoring, applying the zero product property, and verifying solutions is essential for solving quadratic equations effectively. This foundational skill supports deeper understanding and application in various mathematical and real-world contexts.

Solving Quadratic Equations by Factoring Example 1

To solve a quadratic equation by factoring, the first step is to write the quadratic in standard form, which means arranging all terms on one side of the equation so that the other side equals zero. For example, if the equation is not already in this form, you can subtract or add terms on both sides to achieve it. Once in standard form, identify the coefficients a, b, and c from the quadratic expression ax² + bx + c = 0.

When the coefficient a is not equal to one, the ac method is an effective factoring strategy. This involves multiplying a and c to find a product, then determining factor pairs of this product that sum to the middle coefficient b. For instance, if a = -1, b = -3, and c = 4, then ac = -4. The factor pairs of -4 are (1, -4) and (-1, 4). The pair that sums to b = -3 is (1, -4).

Next, rewrite the middle term as the sum of these two factors, effectively breaking it apart. This transforms the quadratic into four terms, which can then be grouped into two pairs. Factoring out the greatest common factor (GCF) from each pair reveals a common binomial factor. For example, grouping terms might yield expressions like x(-x + 1) and 4(-x + 1), where -x + 1 is the common binomial.

Factoring out this common binomial results in the product of two binomials, such as (-x + 1)(x + 4) = 0. Setting each factor equal to zero allows solving for the variable x. For -x + 1 = 0, adding x to both sides gives x = 1. For x + 4 = 0, subtracting 4 from both sides gives x = -4. These values are the solutions to the quadratic equation.

To verify the solutions, substitute each value back into the original equation to ensure the equation holds true. This process confirms the accuracy of the factoring and solution steps.

Use the zero product rule to solve the following equations for $x$.
$(x-5)(x+3)=0$

• A. $x=-5$ or $x=-3$
• B. $x=5$ or $x=3$
• C. $x=-5$ or $x=3$
• D. $x=5$ or $x=-3$

Answer: (D)

Use the zero product rule to solve the following equations for $x$.
$2(x-4)(x+6)=0$

• A. $x=4$ or $x=-6$
• B. $x=-4$ or $x=6$
• C. $x=4$ or $x=6$
• D. $x=-4$ or $x=-6$

Answer: (A)

Solve the following quadratic equations.
$2x^2-10x=0$

• A. $x=0$ or $x=-5$
• B. $x=2$ or $x=-5$
• C. $x=0$ or $x=5$
• D. $x=0$ or $x=10$

Answer: (C)

Solve the following quadratic equations.
$2x^2=5x+3$

• A. $x=\frac12$ or $x=-3$
• B. $x=-\frac12$ or $x=3$
• C. $x=-\frac12$ or $x=-3$
• D. $x=\frac12$ or $x=3$

Answer: (B)

Here’s what students ask on this topic:

What is the standard form of a quadratic equation and why is it important?

The standard form of a quadratic equation is $a{x}^2+bx+c=0$, where $a$, $b$, and $c$ are constants and $a$ ≠ 0. This form is important because it provides a clear structure to solve the equation by factoring, completing the square, or using the quadratic formula. Writing the quadratic in standard form allows us to identify coefficients and apply methods systematically to find the values of $x$ that satisfy the equation. It also helps in modeling real-world problems where relationships are quadratic, such as projectile motion or business profit analysis.

How do you factor a quadratic equation to solve for x?

To factor a quadratic equation like $a{x}^2+bx+c=0$, first ensure it is in standard form. Then, find two numbers that multiply to $c$ and add to $b$. These numbers help break the middle term or directly factor the trinomial into two binomials, for example, $(x+\mathrm{m})(x+\mathrm{n})$. After factoring, apply the zero product property by setting each binomial equal to zero: $x+\mathrm{m}=\; 0$ and $x+\mathrm{n}=\; 0$. Solving these gives the values of $x$ that satisfy the equation.

What is the zero product property and how is it used in solving quadratic equations?

The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. In mathematical terms, if $AtimesB=\; 0$, then either $A=\; 0$ or $B=\; 0$. This property is crucial when solving quadratic equations after factoring. Once the quadratic is factored into binomials, such as $(x+\mathrm{m})(x+\mathrm{n})\; =\; 0$, we set each factor equal to zero separately. Solving these simple linear equations gives the roots or solutions of the quadratic equation.

How can you verify the solutions of a quadratic equation after factoring?

After finding the solutions by factoring, verification involves substituting each solution back into the original quadratic equation to check if it satisfies the equation. For example, if the solutions are $x=\; -3$ and $x=\; -7$, plug these values into $x{x}^2+\; 10x+\; 21\; =\; 0$. Calculate each term and confirm that the sum equals zero. This step ensures the solutions are correct and helps reinforce understanding of the equation's behavior.

What real-world applications can quadratic equations model?

Quadratic equations model many real-world scenarios. For example, they describe the path of projectiles under gravity, such as balls thrown in sports or objects launched into the air. They also model business problems like profit maximization, where revenue and cost functions are quadratic, helping to find optimal pricing or production levels. Additionally, quadratic equations appear in physics, engineering, and biology to represent relationships involving acceleration, area, or growth patterns. Understanding how to solve these equations allows us to analyze and predict outcomes in these contexts.