**Graphing Quadratic Equation and Find the Nature of Roots :**

In this section, you will learn, how to examine the nature of roots of a quadratic equation using its graph.

To obtain the roots of the quadratic equation

ax^{2} + bx + c = 0

graphically, we first draw the graph of

y = ax^{2} +bx +c

The solutions of the quadratic equation are the x coordinates of the points of intersection of the curve with x-axis.

To find the questions i and ii, please visit the page "How to Find Nature of Solution of Quadratic Equation with Graph"

To find the questions iii and iv, please visit the page "Finding Nature of Quadratic Equation by Graphing"

**Question 1 :**

Graph the following quadratic equations and state their nature of solutions.

(v) x^{2} - 6x + 9 = 0

**Solution :**

Draw the graph for the function y = x^{2} - 6x + 9

Let us give some random values of x and find the values of y.

x -4 -3 -2 -1 0 1 2 3 4 |
x 16 9 4 1 0 1 4 9 16 |
-6x -24 -18 -12 -6 0 -6 -12 -18 -24 |
+9 9 9 9 9 9 9 9 9 9 |
y 1 0 1 4 9 4 1 0 1 |

**Points to be plotted :**

(-4, 1) (-3, 0) (-2, 1) (-1, 4) (0, 9) (1, 4) (2, 1) (3, 0) (4, 1)

To find the x-coordinate of the vertex of the parabola, we may use the formula x = -b/2a

x = -(-6)/2(1) = 6/2 = 3

By applying x = 3, we get the value of y.

y = 3^{2} - 6(3) + 9

y = 9 - 18 + 9

y = 0

Vertex (3, 0)

The graph of the given parabola intersect the x-axis at the one point. Hence it has real and equal roots.

(vi) (2x - 3)(x + 2) = 0

**Solution :**

**(2x - 3)(x + 2) = 0**

**2x ^{2} + 4x - 3x - 6 = 0**

**2x ^{2} + x - 6 = 0**

Let us give some random values of x and find the values of y.

y = **2x ^{2} + x - 6**

x -4 -3 -2 -1 0 1 2 3 4 |
2x 32 18 8 2 0 1 8 18 32 |
x -4 -3 -2 -1 0 1 2 3 4 |
-6 -6 -6 -6 -6 -6 -6 -6 -6 -6 |
y 22 9 0 -5 -6 -4 4 15 30 |

**Points to be plotted :**

(-4, 22) (-3, 9) (-2, 0) (-1, -5) (0, -6) (1, -4) (2, 4) (3, 15) (4, 30)

To find the x-coordinate of the vertex of the parabola, we may use the formula x = -b/2a

x = -1/2(2) = 1/4

By applying x = 1/4, we get the value of y.

y = **2(1/4) ^{2} + (1/4) - 6**

y = 2(1/16) + (1/4) - 6

y = -45/8

Vertex (1/4, -45/8)

The graph of the given parabola intersects the x-axis at two points. Hence it has two real and unequal roots.

After having gone through the stuff given above, we hope that the students would have understood, how to examine the nature of roots of a quadratic equation using its graph.

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