Use Factorizing To Solve The Equation Below:$4x^2 - 13x = 35$

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving quadratic equations using factorization, a powerful technique that can be used to find the solutions of quadratic equations. We will use the equation $4x^2 - 13x = 35$ as an example to demonstrate the steps involved in solving quadratic equations using factorization.

What is Factorization?

Factorization is a mathematical technique used to express an algebraic expression as a product of simpler expressions. In the context of quadratic equations, factorization involves expressing the quadratic expression as a product of two binomial expressions. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Step 1: Rearrange the Equation

To solve the equation $4x^2 - 13x = 35$, we need to rearrange it to the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$. We can do this by subtracting 35 from both sides of the equation:

$$4x^2 - 13x - 35 = 0$$

Step 2: Factorize the Quadratic Expression

Now that we have the equation in the standard form, we can try to factorize the quadratic expression. Factorization involves finding two binomial expressions whose product is equal to the quadratic expression. In this case, we can try to factorize the quadratic expression as follows:

$$(2x + ?)(2x + ?) = 4x^2 - 13x - 35$$

We need to find two binomial expressions whose product is equal to the quadratic expression. We can start by trying different combinations of binomial expressions.

Step 3: Find the Factors

After trying different combinations of binomial expressions, we find that the quadratic expression can be factorized as follows:

$$(2x + 7)(2x - 5) = 4x^2 - 13x - 35$$

Step 4: Solve for x

Now that we have factorized the quadratic expression, we can solve for $x$ by setting each factor equal to zero and solving for $x$. We have two factors:

$2x + 7 = 0$ and $2x - 5 = 0$

We can solve for $x$ by subtracting 7 from both sides of the first equation and adding 5 to both sides of the second equation:

2x = -7$ and $2x = 5

Dividing both sides of each equation by 2, we get:

x = -\frac{7}{2}$ and $x = \frac{5}{2}

Conclusion

In this article, we have demonstrated how to solve quadratic equations using factorization. We used the equation $4x^2 - 13x = 35$ as an example to show the steps involved in solving quadratic equations using factorization. We rearranged the equation to the standard form, factorized the quadratic expression, and solved for $x$ by setting each factor equal to zero and solving for $x$. We found two solutions for $x$, which are $x = -\frac{7}{2}$ and $x = \frac{5}{2}$.

Tips and Tricks

• Factorization is a powerful technique for solving quadratic equations, but it may not always be possible to factorize the quadratic expression.
• When factorizing a quadratic expression, it is often helpful to look for common factors or to use the quadratic formula.
• When solving for $x$, make sure to check your solutions by plugging them back into the original equation.

Real-World Applications

Quadratic equations have many real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity or other forces.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
• Economics: Quadratic equations are used to model the behavior of economic systems, such as supply and demand curves.

Conclusion

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In our previous article, we demonstrated how to solve quadratic equations using factorization. In this article, we will provide a Q&A guide to help you better understand quadratic equations and how to solve them.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable (usually x) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I know if a quadratic equation can be solved using factorization?

A: To determine if a quadratic equation can be solved using factorization, you need to check if the quadratic expression can be expressed as a product of two binomial expressions. If it can, then factorization is a viable method for solving the equation.

Q: What are the steps involved in solving a quadratic equation using factorization?

A: The steps involved in solving a quadratic equation using factorization are:

1. Rearrange the equation to the standard form of a quadratic equation.
2. Factorize the quadratic expression.
3. Solve for x by setting each factor equal to zero and solving for x.

Q: What are some common mistakes to avoid when solving quadratic equations using factorization?

A: Some common mistakes to avoid when solving quadratic equations using factorization include:

• Not rearranging the equation to the standard form of a quadratic equation.
• Not factorizing the quadratic expression correctly.
• Not solving for x correctly.

Q: What are some tips and tricks for solving quadratic equations using factorization?

A: Some tips and tricks for solving quadratic equations using factorization include:

• Look for common factors or use the quadratic formula to help factorize the quadratic expression.
• Check your solutions by plugging them back into the original equation.
• Use a calculator or computer software to help solve the equation.

Q: How do I know if a quadratic equation has real or complex solutions?

A: To determine if a quadratic equation has real or complex solutions, you need to check the discriminant of the equation. The discriminant is the expression under the square root in the quadratic formula. If the discriminant is positive, then the equation has two real solutions. If the discriminant is zero, then the equation has one real solution. If the discriminant is negative, then the equation has two complex solutions.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have many real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity or other forces.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
• Economics: Quadratic equations are used to model the behavior of economic systems, such as supply and demand curves.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. We have provided a Q&A guide to help you better understand quadratic equations and how to solve them. We have also discussed the real-world applications of quadratic equations and how they are used in physics, engineering, and economics.

Additional Resources

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Equations
• Wolfram Alpha: Quadratic Equations

Final Thoughts

Quadratic equations are a powerful tool for modeling and solving problems in mathematics and other fields. By understanding how to solve quadratic equations using factorization, you can gain a deeper understanding of the underlying mathematics and develop problem-solving skills that can be applied to a wide range of situations.