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Solve quadratic problems?
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more than 2 weeks ago
To solve a quadratic problem, you can use the quadratic formula or factor the quadratic equation. Here is the general form of a quadratic equation:
ax^2 + bx + c = 0
1. Using the quadratic formula:
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation. To solve the quadratic equation using the quadratic formula, substitute the values of a, b, and c into the formula and solve for x.
2. Factoring the quadratic equation:
If the quadratic equation can be factored, you can factor it into two binomial expressions and set each factor equal to zero to solve for x. For example, if the quadratic equation is:
x^2 + 5x + 6 = 0
You can factor it as:
(x + 2)(x + 3) = 0
Setting each factor equal to zero gives:
x + 2 = 0 or x + 3 = 0
Solving for x gives:
x = -2 or x = -3
These are the two solutions to the quadratic equation.
ax^2 + bx + c = 0
1. Using the quadratic formula:
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation. To solve the quadratic equation using the quadratic formula, substitute the values of a, b, and c into the formula and solve for x.
2. Factoring the quadratic equation:
If the quadratic equation can be factored, you can factor it into two binomial expressions and set each factor equal to zero to solve for x. For example, if the quadratic equation is:
x^2 + 5x + 6 = 0
You can factor it as:
(x + 2)(x + 3) = 0
Setting each factor equal to zero gives:
x + 2 = 0 or x + 3 = 0
Solving for x gives:
x = -2 or x = -3
These are the two solutions to the quadratic equation.
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Google Gemini AI
more than 2 weeks ago
Solving Quadratic Equations
1. Factoring:
- Factor the quadratic equation into the form (ax + b)(cx + d) = 0.
- Set each factor equal to zero and solve for x: ax + b = 0, cx + d = 0.
- The solutions to the equation are the values of x that satisfy either of these factors.
2. Quadratic Formula:
- If the quadratic equation cannot be factored easily, use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
- Where a, b, and c are the coefficients of the quadratic equation (ax² + bx + c = 0).
3. Completing the Square:
- Add and subtract the square of half the coefficient of the linear term (b/2) to the equation.
- Factor the resulting perfect square trinomial.
- Solve for x by taking the square root of both sides.
4. Graphing:
- Graph the quadratic equation y = ax² + bx + c.
- The solutions to the equation are the x-intercepts of the graph.
Example 1: Factoring
Solve the equation: x² - 5x + 6 = 0
- (x - 2)(x - 3) = 0
- x - 2 = 0 or x - 3 = 0
- x = 2 or x = 3
Example 2: Quadratic Formula
Solve the equation: x² + 2x - 3 = 0
- a = 1, b = 2, c = -3
- x = (-2 ± √(2² - 4(1)(-3))) / 2(1)
- x = (-2 ± √16) / 2
- x = (-2 ± 4) / 2
- x = 1 or x = -3
Example 3: Completing the Square
Solve the equation: x² + 4x + 3 = 0
- x² + 4x + 4 - 4 + 3 = 0
- (x + 2)² - 1 = 0
- (x + 2)² = 1
- x + 2 = ±1
- x = -1 or x = -3
1. Factoring:
- Factor the quadratic equation into the form (ax + b)(cx + d) = 0.
- Set each factor equal to zero and solve for x: ax + b = 0, cx + d = 0.
- The solutions to the equation are the values of x that satisfy either of these factors.
2. Quadratic Formula:
- If the quadratic equation cannot be factored easily, use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
- Where a, b, and c are the coefficients of the quadratic equation (ax² + bx + c = 0).
3. Completing the Square:
- Add and subtract the square of half the coefficient of the linear term (b/2) to the equation.
- Factor the resulting perfect square trinomial.
- Solve for x by taking the square root of both sides.
4. Graphing:
- Graph the quadratic equation y = ax² + bx + c.
- The solutions to the equation are the x-intercepts of the graph.
Example 1: Factoring
Solve the equation: x² - 5x + 6 = 0
- (x - 2)(x - 3) = 0
- x - 2 = 0 or x - 3 = 0
- x = 2 or x = 3
Example 2: Quadratic Formula
Solve the equation: x² + 2x - 3 = 0
- a = 1, b = 2, c = -3
- x = (-2 ± √(2² - 4(1)(-3))) / 2(1)
- x = (-2 ± √16) / 2
- x = (-2 ± 4) / 2
- x = 1 or x = -3
Example 3: Completing the Square
Solve the equation: x² + 4x + 3 = 0
- x² + 4x + 4 - 4 + 3 = 0
- (x + 2)² - 1 = 0
- (x + 2)² = 1
- x + 2 = ±1
- x = -1 or x = -3
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