Standards: ⢠Interpret structure of expressions. ⢠MGSE9â12.A.SSE.2 Use the structure of an expression to rewrite it in different equivalent for...
f(x) 2 x 5x 11 and g(x) 4 x 4 x 5 Find 2f(x) + 3g(x) Find g(x) – f(x) Find f(-3)
Solving Quadratic Equations
• Standards: • Interpret structure of expressions • MGSE9‐12.A.SSE.2 Use the structure of an expression to rewrite it in different equivalent forms. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2) (x2 + y2). • • Write expressions in equivalent forms to solve problems • MGSE9–12.A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. • • MGSE9–12.A.SSE.3a Factor any quadratic expression to reveal the zeros of the function defined by the expression. • • MGSE9–12.A.SSE.3b Complete the square in a quadratic expression to reveal the maximum and minimum value of the function defined by the expression.
• Learning Target:
•Interpret the structure of expressions. (Quadratics)
What is a Quadratic Equation? A QUADRATIC EQUATION is an equation in which the greatest power of any variable is 2.
The standard form of a quadratic equation is 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 , where a, b, and c are real numbers and a ≠ 0.
Quadratic Equations The factors of a quadratic equation in standard form are related to the x-intercepts of the graph of its related function. Therefore, you can find or confirm the factors of a polynomial by looking at the xintercepts of the graph of its related function.
2
Factor 𝑦 = 𝑥 + 3𝑥 − 10. Then plot the x-intercepts using a calculator. There is no GCF so continue with factoring the trinomial. 𝑦 = 𝑥 2 − 2𝑥 + 5x − 10 (𝑥 2 − 2𝑥) + (5x − 10) = 0
𝑥 𝑥−2 +5 𝑥−2 =0
𝒙−𝟐 𝒙+𝟓 =𝟎
Use the graphing calculator to plot at least 4 points. Where are the xintercepts located?
(-5, 0) and (2, 0) What is the relation between the factored form and the xintercepts?
The factors, when solved for zero, give the location of the xintercepts.
The factors, when solved for zero, give the location of the x-intercepts. To find the solution to the quadratic equation, 1)factor, 2)set each factor equal to zero, then 3)solve each factor for x. The solution 1) 𝑥 2 − 2𝑥 + 5x − 10 = 0 to the quadratic 𝑥−2 𝑥+5 =0 equation is 2) 𝑥 − 2 = 0 and (𝑥 + 5) = 0 {−𝟓, 𝟐}
