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Questions and Answers
What are the solutions to the equation $x^2 - 4x + 4 = 0$ when solved using the Quadratic formula?
What are the solutions to the equation $x^2 - 4x + 4 = 0$ when solved using the Quadratic formula?
- x = -2
- x = 2 (correct)
- x = 4
- x = 0
What is the first step in solving the equation $2x^2 + 14x = 50$ using the Quadratic formula?
What is the first step in solving the equation $2x^2 + 14x = 50$ using the Quadratic formula?
- Combine like terms
- Factor the left side
- Convert the equation into standard form (correct)
- Isolate the x variable
When factoring the equation $x^2 + 12x + 35 = 0$, what products do you look for?
When factoring the equation $x^2 + 12x + 35 = 0$, what products do you look for?
- Two numbers that add to 35 and multiply to -12
- Two numbers that add to 12 and multiply to 35 (correct)
- Two numbers that add to -12 and multiply to 35
- Two numbers that multiply to 12 and add to 35
What are the zeros of the function $h(x) = x^2 + 8x + 12$?
What are the zeros of the function $h(x) = x^2 + 8x + 12$?
What is the product of $(5 + 6i)(-4 + 7i)$?
What is the product of $(5 + 6i)(-4 + 7i)$?
What is the square root of $-24$?
What is the square root of $-24$?
What is the vertex form of the equation $y = x² - 4x - 1$?
What is the vertex form of the equation $y = x² - 4x - 1$?
What is the first step to solve the equation $x² + 4x + 6 = 0$ by completing the square?
What is the first step to solve the equation $x² + 4x + 6 = 0$ by completing the square?
How many real solutions does the equation $x² + 12x + 36 = 0$ have?
How many real solutions does the equation $x² + 12x + 36 = 0$ have?
Using the discriminant, which equation will have two imaginary solutions?
Using the discriminant, which equation will have two imaginary solutions?
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Study Notes
Quadratic Formula
- The quadratic formula solves for x in equations of the form ax² + bx + c = 0.
- The formula is: x = (-b ± √(b² - 4ac)) / 2a
- Substitute a, b, and c values from the equation to solve.
Factoring Quadratics
- Factoring is a method for solving quadratic equations.
- Factor the quadratic expression into two binomials.
- Set each binomial equal to zero and solve for x.
Finding Zeros of a Function
- The zeros of a function are the x-values where the function equals zero.
- To find the zeros, set the function equal to zero and solve for x.
Complex Numbers
- Complex numbers are numbers that include the imaginary unit i.
- i is defined as the square root of -1.
- Complex numbers are written in the form a + bi, where a and b are real numbers.
- Multiplying complex numbers:
- Use the distributive property to multiply the terms.
- Remember that i² = -1.
- Simplify the result.
Vertex Form of a Quadratic Equation
- The vertex form of a quadratic equation is y = a(x - h)² + k.
- The vertex of the parabola is at the point (h, k).
- To write an equation in vertex form, complete the square.
- Completing the square:
- Move the constant term to the right side of the equation.
- Take half of the coefficient of the x term, square it, and add it to both sides of the equation.
- Factor the left side of the equation as a perfect square trinomial.
- Rewrite the equation in vertex form.
Discriminant
- The discriminant of a quadratic equation is the expression b² - 4ac.
- The discriminant can be used to determine the number and type of solutions of the equation.
- Number of solutions:
- If the discriminant is positive, there are two real solutions.
- If the discriminant is zero, there is one real solution.
- If the discriminant is negative, there are two imaginary solutions.
- Type of solutions:
- If the discriminant is a perfect square, the solutions are rational.
- If the discriminant is not a perfect square, the solutions are irrational.
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