16.2 Finding Solutions to Quadratics

16.2 Finding Solutions to Quadratics

Vocabulary

Zeros

Zeros, Roots, Solutions, X-Intercepts

The values of X when Y equals zero. The place where the ball hits the ground.

Hotspot

Mark the zeros.

The Quadratic Formula

A formula to find the zeros of the parabola.

There will usually be two answers and occasionally one. If you wind up with a negative in the square root, there are no zeros.

Labeling

Drag labels to their correct position on the image

b

ac

a

​Example 1:

Multiple Choice

Step 1: Factor $$x^2+10x+21=0$$

• $$\left(x+3\right)\left(x+7\right)$$

• $$\left(x+1\right)\left(x+21\right)$$

• $$\left(x-7\right)\left(x-3\right)$$

• $$\left(x-21\right)\left(x-1\right)$$

Dropdown

Step 2: Set both factors equal to zero: $$\left(x+3\right)=$$ ​

Select answer

Dropdown

Step 2: Set both factors equal to zero: $$\left(x+7\right)=$$ ​

Select answer

Multiple Choice

Step 3: Solve both factors for x: $$x+3=0$$ ​​

What should you do to isolate x?

• Add 0 to both sides

• Subtract 0 from both sides

• Add 3 to both sides

• Subtract 3 from both sides

Drag and Drop

$$x+3=0\ and\ x+7=0$$

x=​

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and x=​

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Multiple Choice

$$x^2+x-20=0$$ is factored into $$\left(x+5\right)\left(x-4\right)=0$$ , what are the zeros?

• $$x=5\ and\ -4$$

• $$x=-5\ and\ -4$$

• $$x=5\ and\ 4$$

• $$x=-5\ and\ 4$$

​Example 2:

Multiple Choice

Step 1: Factor $$3x^2-5x-2=0$$

• $$\left(3x+1\right)\left(x-2\right)$$

• $$\left(3x-1\right)\left(x+2\right)$$

• $$\left(x-2\right)\left(x+1\right)$$

• $$\left(3x+3\right)\left(x-1\right)$$

Dropdown

Step 2: Set both factors equal to zero: $$\left(3x+1\right)=$$ ​

Select answer

Dropdown

Step 2: Set both factors equal to zero: $$\left(x-2\right)=$$ ​

Select answer

Multiple Choice

Step 3: Solve both factors for x: $$3x+1=0$$ ​​

What should you do to isolate x?

• Add 1 to both sides then divide by 3 to make $$\frac{1}{3}$$

• Subtract 1 from both sides then divide by 3 to make $$-\frac{1}{3}$$

• Add 1 to both sides and multiply by 3 to make 3.

• Subtract 1 from both sides and multiply by 3 to make -3.

Drag and Drop

$$3x+1=0\ and\ x-2=0$$

x=​ ​

Placeholder

and x=​ ​

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​

Drag these tiles and drop them in the correct blank above

Multiple Choice

$$2x^2-5x-12=0$$ is factored into $$\left(2x+3\right)\left(x-4\right)=0$$ , what are the zeros?

• $$x=\frac{3}{2}\ and\ -4$$

• $$x=-\frac{3}{2}\ and\ -4$$

• $$x=-\frac{3}{2}\ and\ 4$$

• $$x=-\frac{2}{3}\ and\ 4$$

Quadratic Formula

​Example 3:

Drag and Drop

Given $$2x^2+4x-96=0$$ , a=​

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, b=​

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and c=​

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Drag these tiles and drop them in the correct blank above

Labeling

Given $$2x^2+4x-96=0$$ has $$a=2$$ , $$b=4$$ and $$c=-96$$ , place the numbers in the correct place in the formula.

Drag labels to their correct position on the image

Multiple Choice

Given $$2x^2+4x-96=0$$ , choose the correct quadratic formula.

• $$\frac{4\pm\sqrt[]{4^2-4\left(2\right)\left(-96\right)}}{2\left(2\right)}$$

• $$\frac{-4\pm\sqrt[]{4^2-4\left(2\right)\left(-96\right)}}{2\left(2\right)}$$

• $$\frac{-4\pm\sqrt[]{4-4\left(2\right)\left(-96\right)}}{2\left(2\right)}$$

• $$\frac{-4\pm\sqrt[]{4^2-4\left(2\right)\left(96\right)}}{2\left(2\right)}$$

Dropdown

Given $$\frac{-4\pm\sqrt[]{4^2-4\left(2\right)\left(-96\right)}}{2\left(2\right)}$$ , simplify the function: $$4^2$$ becomes ​

Select answer

, $$-4\left(2\right)\left(-96\right)$$ becomes ​

Select answer

and $$2\left(2\right)$$ becomes ​

Select answer

Multiple Choice

Now $$\frac{-4\pm\sqrt[]{4^2-4\left(2\right)\left(-96\right)}}{2\left(2\right)}$$ has become:

• $$\frac{4\pm\sqrt[]{16+768}}{4}$$

• $$\frac{4\pm\sqrt[]{16-768}}{4}$$

• $$\frac{-4\pm\sqrt[]{16+768}}{4}$$

• $$\frac{-4\pm\sqrt[]{16-768}}{4}$$

Multiple Choice

Since we have $$\frac{-4\pm\sqrt[]{16+768}}{4}$$ and 16+768 creates $$\frac{-4\pm\sqrt[]{784}}{4}$$ , what is $$\sqrt[]{784}$$ ?

• $$\pm27$$

• $$\pm28$$

• $$\pm29$$

• $$\pm30$$

Multiple Choice

Now that we have $$\frac{-4\pm28}{4}$$ , what two factions does this make?

• $$\frac{-4+28}{4}and\ \frac{4+28}{4}$$

• $$\frac{-4+28}{4}and\ \frac{4-28}{4}$$

• $$\frac{4+28}{4}and\ \frac{4-28}{4}$$

• $$\frac{-4+28}{4}and\ \frac{-4-28}{4}$$

Multiple Choice

Since our two fractions are $$\frac{-4+28}{4}and\ \frac{-4-28}{4}$$ , what does they equal?

• $$-6\ and\ 8$$

• $$6\ and\ -8$$

• $$6\ and\ 8$$

• $$-6\ and\ -8$$

Dropdown

Given $$12x^2-12x-6=0$$ , a=​

Select answer

, b=​ ​

Select answer

and c=​

Select answer

Match

Match the factor with the solution:

• $$x=-\frac{1}{2}$$

• $$x=\frac{1}{2}$$

• $$x=2$$

• $$x=-2$$

• $$x=-\frac{2}{3}$$

• $$\left(2x+1\right)=0$$

• $$\left(2x-1\right)=0$$

• $$\left(x-2\right)=0$$

• $$\left(x+2\right)=0$$

• $$\left(3x+2\right)=0$$