Calculate Points A and B on the Quadratic Graph of x² - 6x + 8

Q: Why are points A and B on the x-axis?

Points A and B are x-intercepts - where the parabola crosses the x-axis. At these points, the y-coordinate is always 0, so we write them as (x, 0).

Q: How do I know which numbers to use for factoring?

Look for two numbers that multiply to give 8 (the constant term) and add to give -6 (the x coefficient). Here: -2 × -4 = 8 and -2 + (-4) = -6.

Q: What if I can't factor the quadratic easily?

You can always use the quadratic formula: $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $. For this problem: a = 1, b = -6, c = 8.

Q: Which point is A and which is B?

From the graph, point A appears to be at (2, 0) and point B at (4, 0). The order doesn't affect the mathematics - both are correct x-intercepts!

Q: How can I check my factoring is correct?

Expand your factors! $ (x - 2)(x - 4) = x^2 - 4x - 2x + 8 = x^2 - 6x + 8 $ ✓ This matches the original function.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's find the coordinates of points A and B.

00:12 Notice that A and B are where the graph meets the X-axis.

00:25 At these points, the Y value is zero.

00:30 So, we set Y to zero and solve for X.

00:36 Next, factor the equation into a trinomial.

00:40 Here is the trinomial we get.

00:43 Now, find the zeros of each factor.

00:46 This gives us one solution.

00:50 And here's the second solution.

00:55 So, both solutions answer the question. Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

The following function has been graphed below:

$f(x)=x^2-6x+8$

Calculate points A and B.

2

Step-by-step solution

To solve the problem of finding points A and B on the graph of the function $f(x) = x^2 - 6x + 8$ , we need to determine where this quadratic function equals zero.

Step-by-step Approach:

Step 1: Check Factoring Possibility
The quadratic function is given by:

f(x) = x^2 - 6x + 8

We will attempt to factor this quadratic expression. We are looking for two numbers that multiply to the constant term, 8, and add to the coefficient of $x$ , which is $-6$ .

Step 2: Identify Factors
The numbers $4$ and $2$ multiply to $8$ and add to $-6$ , if we consider their negative counterparts, $-4$ and $-2$ :

(x - 4)(x - 2) = x^2 - 6x + 8

This matches our expression, confirming that it is the correct factorization.

Step 3: Solve for Roots
Set each factor equal to zero:

x - 4 = 0 \quad \Rightarrow \quad x = 4

x - 2 = 0 \quad \Rightarrow \quad x = 2

Thus, the points where the function intersects the x-axis, which are the roots, are $(2,0)$ and $(4,0)$ .

Therefore, the solution to the problem is that points A and B are at $(2,0)$ and $(4,0)$ .

Final Solution:
The points A and B are $(2,0)$ and $(4,0)$ .

3

Final Answer

$(2,0),(4,0)$

Key Points to Remember

Essential concepts to master this topic

X-intercepts: Points where the quadratic function equals zero
Factoring: $x^2 - 6x + 8 = (x - 2)(x - 4)$
Verification: Substitute x = 2 and x = 4: both give f(x) = 0 ✓

Common Mistakes

Avoid these frequent errors

Confusing x-intercepts with y-intercepts
Don't look for where x = 0 when finding points A and B = gives you (0, 8) instead! X-intercepts occur where the graph crosses the x-axis, meaning f(x) = 0. Always set the quadratic equal to zero and solve for x.

Practice Quiz

Test your knowledge with interactive questions

The following function has been graphed below:

$$ f(x)=-x^2+5x+6 $$

Calculate points A and B.

FAQ

Everything you need to know about this question

Why are points A and B on the x-axis?

+

Points A and B are x-intercepts - where the parabola crosses the x-axis. At these points, the y-coordinate is always 0, so we write them as (x, 0).

How do I know which numbers to use for factoring?

+

Look for two numbers that multiply to give 8 (the constant term) and add to give -6 (the x coefficient). Here: -2 × -4 = 8 and -2 + (-4) = -6.

What if I can't factor the quadratic easily?

+

You can always use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . For this problem: a = 1, b = -6, c = 8.

Which point is A and which is B?

+

From the graph, point A appears to be at (2, 0) and point B at (4, 0). The order doesn't affect the mathematics - both are correct x-intercepts!

How can I check my factoring is correct?

+

Expand your factors! $(x - 2)(x - 4) = x^2 - 4x - 2x + 8 = x^2 - 6x + 8$ ✓ This matches the original function.

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Calculate Points A and B on the Quadratic Graph of x² - 6x + 8

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