Solve The Equation:${ x^2 + 41 = 12x + 9 } 11. S O L V E T H E E Q U A T I O N : 11. Solve The Equation: 11. S O L V E T H Ee Q U A T I O N : ${ x^2 - X - 22 = X - 7 } 10. S O L V E T H E E Q U A T I O N : 10. Solve The Equation: 10. S O L V E T H Ee Q U A T I O N : ${ x^2 + 8x - 12 = 21 } 12. S O L V E T H E E Q U A T I O N : 12. Solve The Equation: 12. S O L V E T H Ee Q U A T I O N : ${ x^2 + 10x + 39 = 23 }$13. Solve
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 explore four different quadratic equations and provide step-by-step solutions to each of them.
Equation 1: x^2 + 41 = 12x + 9
Step 1: Rearrange the Equation
The first step in solving a quadratic equation is to rearrange it in the standard form, ax^2 + bx + c = 0. To do this, we need to move all the terms to one side of the equation.
x^2 + 41 - 12x - 9 = 0
Step 2: Combine Like Terms
Next, we need to combine like terms to simplify the equation.
x^2 - 12x + 32 = 0
Step 3: Factor the Equation
Now, we need to factor the equation to find the values of x. Unfortunately, this equation does not factor easily, so we will need to use the quadratic formula.
Step 4: Use the Quadratic Formula
The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = -12, and c = 32. Plugging these values into the formula, we get:
x = (12 ± √((-12)^2 - 4(1)(32))) / 2(1) x = (12 ± √(144 - 128)) / 2 x = (12 ± √16) / 2 x = (12 ± 4) / 2
Step 5: Solve for x
Now, we can solve for x by simplifying the equation.
x = (12 + 4) / 2 or x = (12 - 4) / 2 x = 16 / 2 or x = 8 / 2 x = 8 or x = 4
Therefore, the solutions to the equation x^2 + 41 = 12x + 9 are x = 8 and x = 4.
Equation 2: x^2 - x - 22 = x - 7
Step 1: Rearrange the Equation
The first step in solving a quadratic equation is to rearrange it in the standard form, ax^2 + bx + c = 0. To do this, we need to move all the terms to one side of the equation.
x^2 - x - 22 - x + 7 = 0
Step 2: Combine Like Terms
Next, we need to combine like terms to simplify the equation.
x^2 - 2x - 15 = 0
Step 3: Factor the Equation
Now, we need to factor the equation to find the values of x. Unfortunately, this equation does not factor easily, so we will need to use the quadratic formula.
Step 4: Use the Quadratic Formula
The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = -2, and c = -15. Plugging these values into the formula, we get:
x = (2 ± √((-2)^2 - 4(1)(-15))) / 2(1) x = (2 ± √(4 + 60)) / 2 x = (2 ± √64) / 2 x = (2 ± 8) / 2
Step 5: Solve for x
Now, we can solve for x by simplifying the equation.
x = (2 + 8) / 2 or x = (2 - 8) / 2 x = 10 / 2 or x = -6 / 2 x = 5 or x = -3
Therefore, the solutions to the equation x^2 - x - 22 = x - 7 are x = 5 and x = -3.
Equation 3: x^2 + 8x - 12 = 21
Step 1: Rearrange the Equation
The first step in solving a quadratic equation is to rearrange it in the standard form, ax^2 + bx + c = 0. To do this, we need to move all the terms to one side of the equation.
x^2 + 8x - 12 - 21 = 0
Step 2: Combine Like Terms
Next, we need to combine like terms to simplify the equation.
x^2 + 8x - 33 = 0
Step 3: Factor the Equation
Now, we need to factor the equation to find the values of x. Unfortunately, this equation does not factor easily, so we will need to use the quadratic formula.
Step 4: Use the Quadratic Formula
The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = 8, and c = -33. Plugging these values into the formula, we get:
x = (-8 ± √((8)^2 - 4(1)(-33))) / 2(1) x = (-8 ± √(64 + 132)) / 2 x = (-8 ± √196) / 2 x = (-8 ± 14) / 2
Step 5: Solve for x
Now, we can solve for x by simplifying the equation.
x = (-8 + 14) / 2 or x = (-8 - 14) / 2 x = 6 / 2 or x = -22 / 2 x = 3 or x = -11
Therefore, the solutions to the equation x^2 + 8x - 12 = 21 are x = 3 and x = -11.
Equation 4: x^2 + 10x + 39 = 23
Step 1: Rearrange the Equation
The first step in solving a quadratic equation is to rearrange it in the standard form, ax^2 + bx + c = 0. To do this, we need to move all the terms to one side of the equation.
x^2 + 10x + 39 - 23 = 0
Step 2: Combine Like Terms
Next, we need to combine like terms to simplify the equation.
x^2 + 10x + 16 = 0
Step 3: Factor the Equation
Now, we need to factor the equation to find the values of x. Unfortunately, this equation does not factor easily, so we will need to use the quadratic formula.
Step 4: Use the Quadratic Formula
The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = 10, and c = 16. Plugging these values into the formula, we get:
x = (-10 ± √((10)^2 - 4(1)(16))) / 2(1) x = (-10 ± √(100 - 64)) / 2 x = (-10 ± √36) / 2 x = (-10 ± 6) / 2
Step 5: Solve for x
Now, we can solve for x by simplifying the equation.
x = (-10 + 6) / 2 or x = (-10 - 6) / 2 x = -4 / 2 or x = -16 / 2 x = -2 or x = -8
Therefore, the solutions to the equation x^2 + 10x + 39 = 23 are x = -2 and x = -8.
Conclusion
Quadratic equations can be a challenging topic for many students and professionals. In this article, we will answer some of the most frequently asked questions about quadratic equations.
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. It is written in the form ax^2 + bx + c = 0, where a, b, and c are constants.
Q: How do I solve a quadratic equation?
A: There are several ways to solve a quadratic equation, including factoring, using the quadratic formula, and graphing. The method you choose will depend on the specific equation and your personal preference.
Q: What is the quadratic formula?
A: The quadratic formula is a mathematical formula that can be used to solve quadratic equations. It is written as:
x = (-b ± √(b^2 - 4ac)) / 2a
This formula can be used to find the solutions to a quadratic equation when it cannot be factored easily.
Q: What is the difference between a quadratic equation and a linear equation?
A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a squared variable (x^2), while a linear equation does not.
Q: Can I use a calculator to solve a quadratic equation?
A: Yes, you can use a calculator to solve a quadratic equation. Most calculators have a built-in quadratic formula function that can be used to find the solutions to a quadratic equation.
Q: How do I determine if a quadratic equation has real or complex solutions?
A: To determine if a quadratic equation has real or complex solutions, you can use the discriminant (b^2 - 4ac). If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has two complex solutions.
Q: Can I use a graphing calculator to solve a quadratic equation?
A: Yes, you can use a graphing calculator to solve a quadratic equation. By graphing the equation, you can find the x-intercepts, which represent the solutions to the equation.
Q: What are some common mistakes to avoid when solving quadratic equations?
A: Some common mistakes to avoid when solving quadratic equations include:
- Not following the order of operations (PEMDAS)
- Not simplifying the equation before solving it
- Not using the correct formula or method
- Not checking the solutions to make sure they are valid
Q: How do I check my solutions to a quadratic equation?
A: To check your solutions to a quadratic equation, you can plug them back into the original equation and make sure they are true. You can also use a calculator to check the solutions.
Conclusion
Quadratic equations can be a challenging topic, but with practice and patience, you can become proficient in solving them. By following the steps outlined in this article and using the quadratic formula, you can solve any quadratic equation that comes your way. Remember to check your solutions and avoid common mistakes to ensure that you are getting accurate results.