NCERT Solutions Class 10 Maths Chapter 4 — Quadratic Equations

Solution:

Step 1: Option A: (x + 1)² = 2x + 3. Expanding the left side gives x² + 2x + 1 = 2x + 3. Simplifying, we get x² - 2 = 0, which is a quadratic equation (a=1, b=0, c=-2).

Step 2: Option B: x(x + 2) = x² + 5. Expanding gives x² + 2x = x² + 5. Simplifying, we get 2x - 5 = 0, which is a linear equation.

Step 3: Option C: x³ - 4x² + 5 = 0. This is a cubic equation, not quadratic, as the highest power of x is 3.

Step 4: Option D: (x - 2)(x + 2) = x² - 4. Expanding gives x² - 4 = x² - 4. Simplifying, we get 0 = 0, which is an identity and not an equation in x.

Solution:

Step 1: By definition, a 'root' or 'solution' of an equation is a value that, when substituted for the variable (x in this case), makes the equation true.

Step 2: Therefore, if 'p' is a root of ax² + bx + c = 0, then substituting x = p into the equation must result in ap² + bp + c = 0.

Step 3: The other options are either specific conditions (like sum of coefficients, formula for equal roots, or condition for distinct real roots) and not generally true for any root 'p'.

Solution:

Step 1: For the equation x² - 7x + 12 = 0, we need to find two numbers whose product is 12 (coefficient of x² × constant term) and whose sum is -7 (coefficient of x).

Step 2: The numbers -3 and -4 satisfy these conditions: (-3) × (-4) = 12 and (-3) + (-4) = -7.

Step 3: Ravi's split, x² - 3x - 4x + 12 = 0, is therefore correct. The next logical step in factorization would be to group terms: x(x-3) - 4(x-3) = 0.

Solution:

Step 1: Let the first positive integer be x. Since the integers are consecutive, the next positive integer will be x + 1.

Step 2: The problem states that their product is 306. So, we can write the equation: x(x + 1) = 306.

Step 3: Expand the left side: x² + x = 306.

Step 4: To form a standard quadratic equation (ax² + bx + c = 0), bring all terms to one side: x² + x - 306 = 0.

Solution:

Step 1: For an expression of the form x² + bx, to complete the square, we need to add (b/2)².

Step 2: In the given expression x² - 10x, the coefficient of x is b = -10.

Step 3: So, we need to add (-10/2)² = (-5)² = 25.

Step 4: Thus, x² - 10x + 25 can be written as (x - 5)².