2. Complete the square to determine the vertex and axis of symmetry of y=−x2+8x+7

Question

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3. Find the zeros of the quadratic equation y=6x2+13x+5
. You may use any of the following methods: factoring, square roots, complete the square, or quadratic formula.

Please show all steps to solve these two questions. Upload a picture of your work for each and clearly label them as #2 and #3. You may use Desmos to check your work, but you must upload your handwritten work to receive credit.

(6 points)

Answer 1

Let's solve both problems step by step.

Problem 2: Complete the square for $y=-x^2+8x+7$

Step 1: Factor out the negative sign from the quadratic terms

$$y=-(x^2-8x)+7$$

Step 2: Complete the square inside the parentheses

To complete the square, we need to take half of the coefficient of $x$ (which is -8), square it and add and subtract that value inside the parentheses.

Half of -8 is -4. Squaring -4 gives us 16.

So, we rewrite the expression as follows:

$$y=-(x^2-8x+16-16)+7$$

This simplifies to:

$$y=-((x-4{)}^2-16)+7$$

Step 3: Distribute the negative sign

Now, distribute the negative sign and simplify:

$$y=-(x-4{)}^2+16+7$$ $$y=-(x-4{)}^2+23$$

Step 4: Identify the vertex and axis of symmetry

The vertex form of a quadratic function is $y=a(x-h{)}^2+k$, where $(h,k)$ is the vertex.

From our equation, we have:

• Vertex $(h,k)=(4,23)$
• Axis of symmetry: $x=h=4$

Problem 3: Find the zeros of the quadratic equation $y=6x^2+13x+5$

Step 1: Apply the quadratic formula

The quadratic formula is given by:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

For our quadratic $a{x}^2+bx+c$ where $a=6$, $b=13$, and $c=5$:

Step 2: Calculate the discriminant $b^2-4ac$

$$b^2={13}^2=169$$ $$4ac=4(6)(5)=120$$ $$b^2-4ac=169-120=49$$

Step 3: Substitute values into the quadratic formula

Since the discriminant is positive, we have two distinct real roots.

$$x=\frac{-13\pm \sqrt{49}}{2\cdot 6}=\frac{-13\pm 7}{12}$$

Step 4: Calculate the two solutions

1. Using $+$: $$x=\frac{-13+7}{12}=\frac{-6}{12}=-\frac{1}{2}$$

2. Using $-$: $$x=\frac{-13-7}{12}=\frac{-20}{12}=-\frac{5}{3}$$

Final Roots: The zeros are $x=-\frac{1}{2}$ and $x=-\frac{5}{3}$.

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Now for handwriting, you will need to write these steps clearly on paper. You can upload the handwritten work as images labeled as #2 for the vertex and axis of symmetry and #3 for the zeros of the quadratic equation.

If you need further assistance or a summary, feel free to ask!