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In this lesson, we will learn about the vertex of a quadratic equation.

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Now, this is the graph of the quadratic equation, y= -x^2 +2x.

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Notice that, this is the highest point on the graph. Hence, this point is the vertex of a quadratic equation.

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Also, if you have a quadratic graph as shown here, the lowest point, is also the vertex of the quadratic equation, y = 2x^2 -8x +3.

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Now, there is a formula to calculate the x-coordinate of the vertex of a quadratic equation.

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The formula is, x = -b/2a. Now, what are 'a' and 'b'? Let's find out.

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The general equation of a quadratic equation is given as, y = ax^2 +bx, +c. Where, 'a', 'b' and 'c' are the coefficients for each term respectively.

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Let's see an example on using this formula, by using this equation, y = 2x^2 -8x +3.

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To find out the values of 'a', 'b' and 'c', we can rewrite this equation as, y = 2x^2 + (-8)x +3.

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By comparing this equation with the general equation, we can see that, 'a' is equals to 2, 'b' is equals to -8, and 'c' is equals to 3.

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Knowing this, we can substitute 'b', with -8, and substitute 'a', with 2. Now, we can simplify this term.

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This term is the same as, [-(-8)]/[2(2)]. Negative multiply bracket -8, gives 8. 2 multiply with 2, gives 4. 8 divide by 4, gives 2

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Finally, we have x equals to 2. With this, we know that the vertex of the quadratic equation has the x-coordinate of 2.

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Let's go back to the original equation. Now, we can use the x–coordinate to find the y-coordinate of the vertex.

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To do so, we can substitute x with 2. -8 multiply with 2, gives -16. Adding -16 with 3, gives -13.

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Again, we can substitute x with 2. 2^2 is the same as, 2 multiply by 2. Which is equals to 4. Let’s write down this number.

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Let's continue. 2 multiply with 4, gives 8. 8 minus 13, gives -5.

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Hence, the vertex has y-coordinate of -5. This y-coordinate is located here.

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Hence, the vertex of the equation has the coordinates of (2, -5).

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Let's see another example. Consider the quadratic equation, y = -x^2 +2x. The graph for this equation is shown here.

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Now, the vertex is located at the highest point on the graph. Let's find the coordinates of this vertex.

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Again, we start with the formula for the x-coordinate of the vertex of a quadratic equation, x = -b/2a.

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To find the values of 'a' and 'b', we can compare this equation with the general equation, y = ax^2 + bx +c.

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For easier comparison, we can rewrite this quadratic equation as, y = -1x^2 +4x + 0. Hence, we can see that, 'a' is equals to -1, ‘b’ is equals to 2, and 'c' is equals to 0.

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Knowing this, we can substitute 'b' with 2 and substitute 'a' with -1.

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Let's simplify this term. Now, this term is the same as, -2/2(-1). 2 multiply by -1, gives -2. -2 divides by -2, gives 1. Hence, we have, x equals to 1.

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With this, the vertex has the x-coordinate of 1. Let's change back to the original equation.

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Now, we can use this x-coordinate to find the y-coordinate of the vertex. Here’s how. Substitute x with 1. 2 multiply with 1, gives back 2.

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Let's continue. Again, substitute x with 1. 1^2, is the same as, 1 multiply with 1. This gives 1. Let’s write this down.

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Finally, -1 add with 2, gives 1.

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Hence, the y-coordinate of the vertex is 1. This can be seen here.

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Hence, the coordinates of the vertex of this quadratic equation is (1, 1).

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That is all for this lesson. Try out the practice question to test your understanding.