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how do you find the x-intercepts of the graph of y=2x^2+x-10

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2 Answers

y = 0 at the point where the function intersects x-axis. Thus, yiou have to put

                                                2x2 + x - 10 = 0

and solve the equation. It is better to factor the quadratic expression, which gives us

                                               2x2 + x - 10 = (2x+5)(x-2) = 0

We have 2x +5 = 0     and x-2 = 0. Thus, the intercepts are x = -5/2 = - 2.5    and x = 2. 

Hope you did well on your ECA. I thought I would share some more details just in case there are other students who might benefit from your question this summer.

To find the x-intercepts of the graph of y=2x²+x-10:

Do you know what an x-intercept is? In the x,y coordinate system, the x-value is horizontal (across), and the y-value is vertical (up and down). The x-axis is horizontal, and the y-axis is vertical. The x-intercept is where the graph of your equation crosses the x-axis. The x-axis is always located where the y-value is equal to zero, and the y-axis is always located where the x-value is zero. This is always true, so it holds true regardless of your equation. If you plug in zero for the value of y in your equation, then you can solve for x (get x by itself), and this will give you the x-value when y is zero. This is the x-intercept for your equation (where your equation's graph crosses the x-axis).

Plugging in zero for y for your equation gives you: 0=2x²+x-10. Do you know how to factor this equation? In other words, what times what equals 2x²+x-10? This is the reverse of FOIL. FOIL is the method where we multiply two binomials to get a trinomial equation. The reverse of FOIL is what they're asking you to do here, because they want you to start with the trinomial and factor it into the two binomials that you would multiply together.

Do you remember FOIL? (First, Inside, Outside, Last.)

(2x+5)(x-2) = (First is 2x times x)+(Inside is 5 times x)+(Outside is 2x times negative 2)+(Last is 5 times negative 2).

This equals (First is 2x times x= 2x²)+(Inside is 5 times x=5x)+(Outside is 2x times negative 2=negative 4x)+(Last is 5 times negative 2=negative 10). This is 2x²+5x-4x-10. This equals 2x²+x-10. This means that our equation 0=2x²+x-10 factors into 0=(2x+5)(x-2). Zero times any number is zero. So what value can we put in for x so that 2x+5 equals zero? Set 2x+5 equal to zero and solve for x to find the first value for x. 2x+5=0

2x+5-5=0-5

2x=-5

2x divided by 2 = -5 divided by 2

x=-5/2=-2 1/2=-2.5, so -2.5 is the first x-intercept that we found for our equation.

x-2=0; x-2+2=0+2; x=2, so 2 is the second x-intercept we found for our equation.

To review, see where our equation, 2x² + x - 10 = 0, is the same as (2x+5)(x-2)=0; and that when x is -2.5, or when x is 2, (2x+5)(x-2) equals a y-value of zero. To check our answers, first plug in -2.5 for x into our original equation:

2(-2.5)² + (-2.5) - 10 = 0

2(6.25) - 2.5 - 10 = 12.5 - 2.5 - 10 = 10 - 10 = 0, so our answer of -2.5 works, because we know that 10-10=0 is correct.

Now plug in 2 for x into our original equation:

2(2)² + (2) - 10 = 0; 2(4) + (2) -10 = 0; 10-10=0, so our answer of 2 also works.

Note that the graph of the equation y = 2x² + x - 10 is NOT a straight line. The graph of a straight line would have no more than one x-intercept and no more than one y-intercept. The graph of a straight line has no changes in direction. It heads in only one direction and never changes the direction in which it is headed. The highest exponent in our equation y = 2x² + x - 10 is a 2. (The exponent is the little number that is raised up higher when it's printed in the text. In this equation, it is just to the right of the 2x.) This means that you can take the highest exponent, 2, minus 1 which equals 1, so it makes a "U-turn" (changes direction) one time. If the highest exponent of an equation is 3, then 3-1=2 means that its graph would have two changes in direction. Just subtract one from the highest exponent to get the number of changes in direction that the graph of that equation has. The graph of the our equation y = 2x² + x - 10 has one "U-turn," so even before we figured out exactly what the x-intercepts were for our equation, we could have known just by looking at it that it may have two x-intercepts, depending on exactly where the particular equation's graph is situated. Indeed we did end up figuring out that it has two x-intercepts, where x=-2.5 and where x=2. The graph of this equation is a parabola, which is kind of shaped like a narrow "U."

Some of these concepts might remain a bit confusing. It's hard to explain everything exactly right for each and every student, especially without being able to directly point to the graph and the coordinate system and without being able to directly interact with each individual. But those are a few of the advantages to having tutoring help in person. There are some important technical issues not covered here regarding how to solve for x and how to determine graphing by merely looking at an equation.

Have a great day and a great summer.