Solve The Equation: { (2x - 1)(x - 3) = (2x - 3)(x - 1)$}$10. Solve The Equation: { X(2x + 6) = 2(x^2 - 5)$}$12. Solve The Equation: { (2x + 1)(x - 4) + 1(x - 2)^2 = 13x(x + 2)$} 14. S O L V E T H E E Q U A T I O N : \[ 14. Solve The Equation: \[ 14. S O L V E T H Ee Q U A T I O N : \[ X(x - 1)
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 one.
Equation 1: (2x - 1)(x - 3) = (2x - 3)(x - 1)
To solve this equation, we will start by expanding both sides of the equation using the distributive property.
Step 1: Expand the Left Side of the Equation
The left side of the equation is (2x - 1)(x - 3). We can expand this using the distributive property as follows:
(2x - 1)(x - 3) = 2x(x - 3) - 1(x - 3) = 2x^2 - 6x - x + 3 = 2x^2 - 7x + 3
Step 2: Expand the Right Side of the Equation
The right side of the equation is (2x - 3)(x - 1). We can expand this using the distributive property as follows:
(2x - 3)(x - 1) = 2x(x - 1) - 3(x - 1) = 2x^2 - 2x - 3x + 3 = 2x^2 - 5x + 3
Step 3: Set the Two Expressions Equal to Each Other
Now that we have expanded both sides of the equation, we can set them equal to each other:
2x^2 - 7x + 3 = 2x^2 - 5x + 3
Step 4: Subtract 2x^2 from Both Sides of the Equation
To eliminate the quadratic term, we can subtract 2x^2 from both sides of the equation:
-7x + 3 = -5x + 3
Step 5: Subtract 3 from Both Sides of the Equation
To isolate the linear term, we can subtract 3 from both sides of the equation:
-7x = -5x
Step 6: Add 5x to Both Sides of the Equation
To solve for x, we can add 5x to both sides of the equation:
2x = 0
Step 7: Divide Both Sides of the Equation by 2
Finally, we can divide both sides of the equation by 2 to solve for x:
x = 0
Equation 2: x(2x + 6) = 2(x^2 - 5)
To solve this equation, we will start by expanding both sides of the equation using the distributive property.
Step 1: Expand the Left Side of the Equation
The left side of the equation is x(2x + 6). We can expand this using the distributive property as follows:
x(2x + 6) = 2x^2 + 6x
Step 2: Expand the Right Side of the Equation
The right side of the equation is 2(x^2 - 5). We can expand this using the distributive property as follows:
2(x^2 - 5) = 2x^2 - 10
Step 3: Set the Two Expressions Equal to Each Other
Now that we have expanded both sides of the equation, we can set them equal to each other:
2x^2 + 6x = 2x^2 - 10
Step 4: Subtract 2x^2 from Both Sides of the Equation
To eliminate the quadratic term, we can subtract 2x^2 from both sides of the equation:
6x = -10
Step 5: Divide Both Sides of the Equation by 6
To solve for x, we can divide both sides of the equation by 6:
x = -10/6 x = -5/3
Equation 3: (2x + 1)(x - 4) + 1(x - 2)^2 = 13x(x + 2)
To solve this equation, we will start by expanding both sides of the equation using the distributive property.
Step 1: Expand the Left Side of the Equation
The left side of the equation is (2x + 1)(x - 4). We can expand this using the distributive property as follows:
(2x + 1)(x - 4) = 2x(x - 4) + 1(x - 4) = 2x^2 - 8x + x - 4 = 2x^2 - 7x - 4
The left side of the equation also includes 1(x - 2)^2, which we can expand as follows:
1(x - 2)^2 = (x - 2)^2 = x^2 - 4x + 4
Step 2: Combine the Two Expressions on the Left Side of the Equation
Now that we have expanded both parts of the left side of the equation, we can combine them:
2x^2 - 7x - 4 + x^2 - 4x + 4 = 3x^2 - 11x
Step 3: Expand the Right Side of the Equation
The right side of the equation is 13x(x + 2). We can expand this using the distributive property as follows:
13x(x + 2) = 13x^2 + 26x
Step 4: Set the Two Expressions Equal to Each Other
Now that we have expanded both sides of the equation, we can set them equal to each other:
3x^2 - 11x = 13x^2 + 26x
Step 5: Subtract 3x^2 from Both Sides of the Equation
To eliminate the quadratic term, we can subtract 3x^2 from both sides of the equation:
-11x = 10x^2 + 26x
Step 6: Subtract 26x from Both Sides of the Equation
To isolate the linear term, we can subtract 26x from both sides of the equation:
-37x = 10x^2
Step 7: Add 37x to Both Sides of the Equation
To solve for x, we can add 37x to both sides of the equation:
0 = 10x^2 + 37x
Step 8: Factor Out x from the Right Side of the Equation
Now that we have added 37x to both sides of the equation, we can factor out x from the right side of the equation:
0 = x(10x + 37)
Step 9: Set Each Factor Equal to Zero
To solve for x, we can set each factor equal to zero:
x = 0 10x + 37 = 0
Step 10: Solve for x in the Second Equation
To solve for x in the second equation, we can subtract 10x from both sides of the equation:
37 = -10x
Step 11: Divide Both Sides of the Equation by -10
To solve for x, we can divide both sides of the equation by -10:
x = -37/10
Equation 4: x(x - 1) = 2(x^2 - 5)
To solve this equation, we will start by expanding both sides of the equation using the distributive property.
Step 1: Expand the Left Side of the Equation
The left side of the equation is x(x - 1). We can expand this using the distributive property as follows:
x(x - 1) = x^2 - x
Step 2: Expand the Right Side of the Equation
The right side of the equation is 2(x^2 - 5). We can expand this using the distributive property as follows:
2(x^2 - 5) = 2x^2 - 10
Step 3: Set the Two Expressions Equal to Each Other
Now that we have expanded both sides of the equation, we can set them equal to each other:
x^2 - x = 2x^2 - 10
Step 4: Subtract x^2 from Both Sides of the Equation
To eliminate the quadratic term, we can subtract x^2 from both sides of the equation:
- x = x^2 - 10
Step 5: Subtract x^2 from Both Sides of the Equation
To isolate the linear term, we can subtract x^2 from both sides of the equation:
- x - x^2 = -10
Step 6: Factor Out x from the Left Side of the Equation
Now that we have subtracted x^2 from both sides of the equation, we can factor out x from the left side of the equation:
-x(1 + x) = -10
Step 7: Divide Both Sides of the Equation by -1
To solve for x, we can divide both sides of the equation by -1:
x(1 + x) = 10
Step 8: Set Each Factor Equal to Zero
To solve for x, we can set each factor equal to zero:
x = 0 1 + x = 10
Step 9: Solve for x in the Second Equation
In our previous article, we explored four different quadratic equations and provided step-by-step solutions to each one. In this article, we will answer some of the most frequently asked questions about solving quadratic equations.
Q: What is a quadratic equation?
A: A quadratic equation is a polynomial equation of degree two, which means it has a highest power of two. It is typically 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: To solve a quadratic equation, you can use one of the following methods:
- Factoring: If the quadratic expression can be factored into the product of two binomials, you can set each factor equal to zero and solve for x.
- Quadratic formula: If the quadratic expression cannot be factored, you can use the quadratic formula to find the solutions: x = (-b ± √(b^2 - 4ac)) / 2a.
- Graphing: You can also graph the quadratic function and find the x-intercepts, which represent the solutions to the equation.
Q: What is the quadratic formula?
A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation.
Q: How do I use the quadratic formula?
A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. Then, you can simplify the expression and find the solutions.
Q: What is the difference between the two solutions of the quadratic formula?
A: The two solutions of the quadratic formula are given by:
x = (-b + √(b^2 - 4ac)) / 2a x = (-b - √(b^2 - 4ac)) / 2a
The difference between the two solutions is the value of the expression inside the square root, which is given by:
√(b^2 - 4ac)
Q: What is the discriminant?
A: The discriminant is the expression inside the square root in the quadratic formula, which is given by:
b^2 - 4ac
If the discriminant is positive, the quadratic equation has two distinct solutions. If the discriminant is zero, the quadratic equation has one repeated solution. If the discriminant is negative, the quadratic equation has no real solutions.
Q: How do I determine the number of solutions to a quadratic equation?
A: To determine the number of solutions to a quadratic equation, you can use the discriminant. If the discriminant is:
- Positive, the quadratic equation has two distinct solutions.
- Zero, the quadratic equation has one repeated solution.
- Negative, the quadratic equation has no real solutions.
Q: Can I use the quadratic formula to solve a quadratic equation with complex solutions?
A: Yes, you can use the quadratic formula to solve a quadratic equation with complex solutions. However, you need to be careful when simplifying the expression and finding the solutions.
Q: Are there any other methods for solving quadratic equations?
A: Yes, there are other methods for solving quadratic equations, such as:
- Factoring: If the quadratic expression can be factored into the product of two binomials, you can set each factor equal to zero and solve for x.
- Graphing: You can also graph the quadratic function and find the x-intercepts, which represent the solutions to the equation.
- Completing the square: You can also use the method of completing the square to solve a quadratic equation.
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 you can use to find the solutions.
Q: Are there any online resources for solving quadratic equations?
A: Yes, there are many online resources for solving quadratic equations, such as:
- Wolfram Alpha: A powerful online calculator that can solve quadratic equations and other mathematical problems.
- Mathway: An online math problem solver that can help you solve quadratic equations and other mathematical problems.
- Khan Academy: A free online resource that provides video lessons and practice exercises for solving quadratic equations and other mathematical topics.
We hope this Q&A guide has been helpful in answering your questions about solving quadratic equations. If you have any further questions, please don't hesitate to ask.