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 into binomials, then applying the zero product property by setting each factor equal to zero. This process identifies the roots or solutions, such as $x=-3$ and $x=-7$. Understanding coefficients, terms, and factoring techniques is essential for mastering polynomial equations and their applications.

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 equation \(x^2 + 10x + 21 = 0\), 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 gives \((-3)^2 + 10(-3) + 21 = 9 - 30 + 21 = 0\), confirming it as a valid solution. Similarly, for \(x = -7\), substituting yields \((-7)^2 + 10(-7) + 21 = 49 - 70 + 21 = 0\), also confirming its validity.

Mastering the process of factoring and solving quadratic equations is essential, as these equations frequently appear in various mathematical and applied contexts. The key steps include writing the equation in standard form, factoring the quadratic expression, applying the zero product property, solving for \(x\), and verifying the solutions. This method provides a reliable approach to finding the roots of quadratic equations efficiently.

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 standard form, you can move terms by adding or subtracting them from both sides. Once the quadratic is in the form \(ax^2 + bx + c = 0\), you can proceed to factor it.

When the leading coefficient \(a\) is not equal to 1, the ac method is an effective factoring strategy. This involves identifying the coefficients \(a\), \(b\), and \(c\) from the quadratic. Then, multiply \(a\) and \(c\) to find the product \(ac\). Next, find two numbers that multiply to \(ac\) and add up to \(b\). These two numbers will help break down the middle term into two terms, allowing you to factor by grouping.

For instance, if \(a = -1\), \(b = -3\), and \(c = 4\), then \(ac = -4\). The factor pairs of \(-4\) are \((1, -4)\) and \((-1, 4)\). Since the sum must be \(-3\), the pair \((1, -4)\) works because \$1 + (-4) = -3\(. Using these numbers, rewrite the quadratic as \)-x^2 + x - 4x + 4 = 0\(.

Next, apply factoring by grouping by grouping the first two terms and the last two terms separately. Factor out the greatest common factor (GCF) from each group. For example, from \)-x^2 + x\(, factor out \)x\( to get \)x(-x + 1)\(, and from \)-4x + 4\(, factor out \)-4\( to get \)-4(x - 1)\(. Notice that the binomials are similar, which allows you to factor them out, resulting in \)(x - 1)(-x + 4) = 0\(.

Finally, set each factor equal to zero and solve for \)x\(. For \)x - 1 = 0\(, solving gives \)x = 1\(. For \)-x + 4 = 0\(, solving gives \)x = 4$. These are the solutions to the quadratic equation. To verify, substitute these values back into the original equation to ensure they satisfy it.

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. $x=\frac{1}{2}$ or $x=-3$
• B. $x=-\frac{1}{2}$ or $x=3$
• C. $x=-\frac12$ or $x=-3$
• D. $x=\frac{1}{2}$ 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 $a$ is not zero. This form is important because it organizes the equation clearly, making it easier to apply methods like factoring, completing the square, or using the quadratic formula to find solutions. Writing the quadratic in standard form ensures that all terms are on one side and the equation equals zero, which is essential for solving the equation by setting factors equal to zero. This form also helps in identifying coefficients and understanding the structure of the quadratic, which is crucial for modeling real-world problems and analyzing the behavior of quadratic functions.

How do you factor a quadratic equation to solve it?

To factor a quadratic equation, first ensure it is in standard form: $a{x}^2+bx+c=0$. Then, find two numbers that multiply to $c$ and add to $b$. For example, in $x^2+10x+21=0$, the numbers 3 and 7 multiply to 21 and add to 10. So, factor as $(x+3)(x+7)=0$. Finally, apply the zero product property by setting each factor equal to zero: $x+3=0$ or $x+7=0$, then solve for $x$. This method finds 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 equals zero, then at least one of the factors must be zero. Mathematically, if $A\cdot B=\; 0$, then either $A=\; 0$ or $B=\; 0$. This property is essential in solving quadratic equations after factoring. Once the quadratic is factored into two binomials, such as $(x+3)(x+7)\; =\; 0$, you set each factor equal to zero: $x+3=\; 0$ or $x+7=\; 0$. Solving these gives the solutions $x=\; -3$ and $x=\; -7$. This method efficiently finds the roots of the quadratic equation.

How can you check if the solutions to a quadratic equation are correct?

To check solutions of a quadratic equation, substitute each solution back into the original equation and verify if the equation holds true. For example, if the quadratic is $x^2+10x+21=0$ and the solutions are $x=\; -3$ and $x=\; -7$, plug each into the equation. For $x=\; -3$: calculate $(-3)^2\; +\; 10(-3)\; +\; 21\; =\; 9\; -\; 30\; +\; 21\; =\; 0$. For $x=\; -7$: calculate $(-7)^2\; +\; 10(-7)\; +\; 21\; =\; 49\; -\; 70\; +\; 21\; =\; 0$. Since both equal zero, the solutions are correct. This step confirms the accuracy of your factoring and solving process.

Why is factoring preferred when solving some quadratic equations?

Factoring is often preferred for solving quadratic equations because it provides a straightforward and efficient way to find solutions when the quadratic trinomial can be easily factored into binomials. It avoids more complex methods like completing the square or using the quadratic formula, saving time and effort. Factoring also reinforces understanding of polynomial structure and the relationship between coefficients and roots. However, factoring works best when the quadratic has rational roots and factors neatly. If factoring is difficult or impossible, other methods may be necessary. Overall, factoring is a valuable skill for quickly solving many quadratic equations encountered in algebra and real-world applications.