 UPSKILL MATH PLUS

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Let us discuss how to determine the nature of the solutions for the given quadratic equations.
Example:
1. Find the nature of the solution of the equation $$y = x^2 - 2x - 3$$.

Solution:

Step 1: Draw the graph of the equation $$y = x^2 - 2x - 3$$.

The table of values for the equation $$y = x^2 - 2x - 3$$ is:

 $$x$$ $$-2$$ $$-1$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$x^2$$ $$4$$ $$1$$ $$0$$ $$1$$ $$4$$ $$9$$ $$16$$ $$2x$$ $$-4$$ $$-2$$ $$0$$ $$2$$ $$4$$ $$6$$ $$8$$ $$3$$ $$3$$ $$3$$ $$3$$ $$3$$ $$3$$ $$3$$ $$3$$ $$y$$ $$5$$ $$0$$ $$-3$$ $$-4$$ $$-3$$ $$0$$ $$5$$

Step 2: Plot the points in the graph using a suitable scale. Step 3: Join the points by a smooth curve.

Step 4: In the graph, observe that the curve intersects the $$X$$ - axis at $$2$$ points $$(-1,0)$$ and $$(3,0)$$. Therefore, the roots of the equation are $$-1$$ and $$3$$.

Since there are two points of intersection with the $$X$$ - axis, the given equation has real and unequal roots.

2. Find the nature of the solutions of the equation $$y = x^2 - 10x + 25$$.

Solution:

Step 1: Draw the graph of the equation $$y = x^2 - 10x + 25$$.

The table of values for the equation $$y = x^2 - 10x + 25$$ is given by:

 $$x$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$x^2$$ $$9$$ $$16$$ $$25$$ $$36$$ $$49$$ $$10x$$ $$30$$ $$40$$ $$50$$ $$60$$ $$70$$ $$25$$ $$25$$ $$25$$ $$25$$ $$25$$ $$25$$ $$y$$ $$4$$ $$1$$ $$0$$ $$1$$ $$4$$

Step 2: Plot the points in the graph. Step 3: Join the points by a smooth curve.

Step 4: Here, the curve meets the $$X$$ - axis at only one point. Therefore, the point of intersection of the parabola with $$X$$ - axis for the given equation is $$(5,0)$$.

Since the point of intersection is only one point with $$X$$ - axis, the given quadratic equation has real and equal roots.