Let us recall the following concepts of algebra:
The square root of a polynomial:
The square root of a polynomial \(p(x)\) is defined as an expression \(q(x)\), when multiplied with itself, results in the given number \(p(x)\).
That is, \(p(x) = q(x) \times q(x)\)
The square root of a polynomial can be determined using two methods.
They are:
1. Factorization method
2. Long division method
Quadratic expression:
The polynomial is an expression of degree \(n\) for the variable \(x\) is of the form \(p(x) = a_0 x^n + a_{1}x^{n-1} + … + a_{n -1}x + a_n\), where \(a_0, a_1, a_2, …, a_n\) are coefficients and \(a_0 \ne 0\).
A quadratic expression is a polynomial of degree \(2\) for the variable \(x\) is of the form \(p(x) = ax^2 + bx + c\), \(a \ne 0\) and \(a\), \(b\) and \(c\) are real numbers.
Zero of a quadratic expression:
If the value of the polynomial \(p(x)\) at \(x = k\) is zero \(p(k) = 0\), then the real number \(k\) is called the zero of the polynomial \(p(x)\).
Quadratic equation:
A quadratic equation in the variable \(x\) is an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\) and \(c\) are numbers, \(a \ne 0\). The degree of the quadratic equation is \(2\).
Roots of a quadratic equation:
The value of \(x\) that makes the expression \(ax^2 + bx + c\) is zero, called the roots of the quadratic equation.
Formation of a quadratic equation:
If \(\alpha\) and \(\beta\) are the roots of a quadratic equation, then the general formula to construct the quadratic equation is \(x^2 - (\alpha + \beta) x + \alpha \beta = 0\).
That is, \( x^2 - (\text{sum of roots}) x + \text{product of roots} = 0\).
Sum of the roots \(=\) \(\alpha + \beta\) \(=\) \(\frac{-b}{a}\)
Product of the roots \(=\) \(\alpha \beta\) \(=\) \(\frac{c}{a}\)
The solution of the quadratic equation is the value of the variable that makes the equation zero. We can say these solutions as roots or zeroes.
There are three methods to solve quadratic equations.
Solve by factorisation method:
Let us see the procedure to solve the quadratic equation by the factorisation method.Step 1: Write the given equation in standard form.
Step 2: Express the middle term as the sum of two terms such that the sum satisfies the middle term, and the product should satisfy the extreme product.
Step 3: Group the expression into two linear factors by taking the common term outside.
Step 4: Now, solve for \(x\) by equating each linear factor to zero. The obtained values of \(x\) are called the roots or zeroes of the equation.
Step 2: Express the middle term as the sum of two terms such that the sum satisfies the middle term, and the product should satisfy the extreme product.
Step 3: Group the expression into two linear factors by taking the common term outside.
Step 4: Now, solve for \(x\) by equating each linear factor to zero. The obtained values of \(x\) are called the roots or zeroes of the equation.
Solving a quadratic equation by completing the square method:
Let us see the procedure to solve the quadratic equation by completing the square.
Step 1: Write the given equation in standard form \(ax^2 + bx + c = 0\).
Step 2: Make sure the coefficient of \(x^2\) is \(a = 1\). If not, make it by dividing the equation by \(a\).
Step 3: Move the constant term to the right-hand side of the equation.
Step 4: Add the square of one-half of the coefficient of \(x\) to both sides. [That is, add \(\left(\frac{b}{2}\right)^2\).]
Step 5: Make the equation a complete square and simplify the right-hand side.
Step 6: Solve for \(x\) by taking the square root on both sides.
Solving a quadratic equation by formula method:
The formula for finding the roots of the quadratic equation \(ax^2 + bx + c = 0\) is:
This formula is known as the quadratic formula.
Nature of roots:
\(b^2 - 4ac\) is called the discriminant of the quadratic equation \(ax^2 + bx + c = 0\). It is denoted by the letter \(\Delta\) or \(D\).
Case I: If \(\Delta = b^2 - 4ac > 0\) then the roots are real and distinct.
Case II: If \(\Delta = b^2 - 4ac = 0\), then the roots are real and equal.
Case III: If \(\Delta = b^2 - 4ac < 0\) then there are no real roots.
Relation between roots and coefficients of a quadratic equation:
If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(ax^2 + bx + c = 0\), then:
(i) Sum of the roots \(=\)
(ii) Product of the roots \(=\)
Procedure to find the solution of quadratic equation graphically:
The steps to find the solution of the quadratic equation graphically is given by:
Step 1: Draw the graph of the equation \(y = ax^2 + bx + c\).
Step 2: (i) If the graph intersects the \(X\) - axis at only one point, the given quadratic equation has only one root or two real and equal roots.
(ii) If the graph intersects the \(X\) - axis at two distinct points, the given quadratic equation has two real and unequal roots.
(iii) If the graph does not intersect the \(X\) - axis at any point, the given equation has no real root.
Step 2: (i) If the graph intersects the \(X\) - axis at only one point, the given quadratic equation has only one root or two real and equal roots.
(ii) If the graph intersects the \(X\) - axis at two distinct points, the given quadratic equation has two real and unequal roots.
(iii) If the graph does not intersect the \(X\) - axis at any point, the given equation has no real root.
Procedure for solving quadratic equations through intersection of lines:
If \(2\) quadratic equations were provided, we need to determine the point of intersection of these two lines.
The steps to find the roots of the intersection point of a quadratic equation are given by:
The steps to find the roots of the intersection point of a quadratic equation are given by:
Step 1: Draw the graph of the quadratic equation, we get a parabola.
Step 2: Now, subtract the \(2\) quadratic equations, and we get an equation of a straight line.
Step 3: Draw the graph of the straight line.
Step 4: (i) If the straight line intersects the parabola at \(2\) distinct points, then the \(x\) - coordinates of the point of intersection will be the roots of the given quadratic equation.
(ii) If the straight line intersects the parabola at only \(1\) point, then the \(x\) - coordinate of the point of intersection will be the root of the given quadratic equation.
(iii) If the straight line does not intersect the parabola, then the given quadratic equation will have no real roots.
Step 2: Now, subtract the \(2\) quadratic equations, and we get an equation of a straight line.
Step 3: Draw the graph of the straight line.
Step 4: (i) If the straight line intersects the parabola at \(2\) distinct points, then the \(x\) - coordinates of the point of intersection will be the roots of the given quadratic equation.
(ii) If the straight line intersects the parabola at only \(1\) point, then the \(x\) - coordinate of the point of intersection will be the root of the given quadratic equation.
(iii) If the straight line does not intersect the parabola, then the given quadratic equation will have no real roots.
Graph of variations:
In a direct variation, the value of one quantity increases(\(\uparrow\))/decreases(\(\downarrow\)), and then the other quantity also increases(\(\uparrow\))/decreases(\(\downarrow\)).
In an indirect variation, the value of one quantity increases(\(\uparrow\))/decreases(\(\downarrow\)), and then the other quantity decreases(\(\downarrow\))/increases(\(\uparrow\)).