Lesson 9

Solving Quadratic Equations by Using Factored Form

View Student Lesson

Problem 1

Find all the solutions to each equation.

  1. \(x(x-1)=0\)
  2. \((5-x)(5+x)=0\)
  3. \((2x+1)(x+8)=0\)
  4. \((3x-3)(3x-3)=0\)
  5. \((7-x)(x+4)=0\)

Solution

Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.

Problem 2

Rewrite each equation in factored form and solve using the zero product property.

  1. \(d^2 -7d+6=0\)
  2. \(x^2 +18x +81=0\)
  3. \(u^2 +7u -60=0\)
  4. \(x^2+0.2x+0.01=0\)

Solution

Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.

Problem 3

Here is how Elena solves the quadratic equation \(x^2 -3x -18 =0\).

\(\displaystyle \begin{align} x^2 -3x -18 &=0\\ (x-3)(x+6)&=0\\ x-3=0 \quad \text { or } &\quad x+6=0\\ x=3\quad \text{ or } &\quad x= \text- 6\\ \end{align}\\\)

Is her work correct? If you think there is an error, explain the error and correct it.

Otherwise, check her solutions by substituting them into the original equation and showing that the equation remains true.

Solution

Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.

Problem 4

Jada is working on solving a quadratic equation, as shown here.

\(\begin{align} p^2-5p&=0\\p(p-5)&=0\\p-5&=0\\p&=5\end{align}\)

She thinks that her solution is correct because substituting 5 for \(p\) in the original expression \(p^2- 5p\) gives \(5^2 - 5(5)\), which is \(25-25\) or 0.

Explain the mistake that Jada made and show the correct solutions.

Solution

Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.

Problem 5

Choose a statement to correctly describe the zero product property. 

If \(a\) and \(b\) are numbers, and \(a \boldcdot b=0\), then:

A:

Both \(a\) and \(b\) must equal 0.

B:

Neither \(a\) nor \(b\) can equal 0.

C:

Either \(a=0\) or \(b=0\).

D:

\(a+b\) must equal 0.

Solution

Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.

(From Unit 7, Lesson 4.)

Problem 6

Which expression is equivalent to \(x^2-7x+12\)?

A:

\((x+3)(x+4)\)

B:

\((x-3)(x-4)\)

C:

\((x+2)(x+6)\)

D:

\((x-2)(x-6)\)

Solution

Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.

(From Unit 7, Lesson 6.)

Problem 7

These quadratic expressions are given in standard form. Rewrite each expression in factored form. If you get stuck, try drawing a diagram.

  1. \(x^2 +7x+6\)
  2. \(x^2 -7x+6\)
  3. \(x^2 -5x+6\)
  4. \(x^2 +5x+6\)

Solution

Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.

(From Unit 7, Lesson 6.)

Problem 8

Select all the functions whose output values will eventually overtake the output values of function \(f\) defined by \(f(x)=25x^2\).

A:

\(g(x)=5(2)^x\)

B:

\(h(x)=5^x\)

C:

\(j(x)=x^2+5\)

D:

\(k(x)=(\frac52)^x\)

E:

\(m(x)=5+2^x\)

F:

\(n(x)=2x^2+5\)

Solution

Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.

(From Unit 6, Lesson 4.)

Problem 9

A piecewise function, \(p\), is defined by this rule: \(p(x)=\begin{cases} x-1, & x\leq \text- 2 \\ 2x-1,& x>\text-2\\ \end{cases} \)

Find the value of \(p\) at each given input.

  1. \(p(\text-20)\)
  2. \(p(\text- 2)\)
  3. \(p(4)\)
  4. \(p(5.7)\)

Solution

Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.

(From Unit 4, Lesson 12.)