{"id":12619,"date":"2024-03-04T10:53:36","date_gmt":"2024-03-04T05:23:36","guid":{"rendered":"https:\/\/tsboardsolutions.in\/?p=12619"},"modified":"2024-03-05T14:22:51","modified_gmt":"2024-03-05T08:52:51","slug":"ts-10th-class-maths-notes-chapter-5","status":"publish","type":"post","link":"https:\/\/tsboardsolutions.in\/ts-10th-class-maths-notes-chapter-5\/","title":{"rendered":"TS 10th Class Maths Notes Chapter 5 Quadratic Equations"},"content":{"rendered":"

We are offering TS 10th Class Maths Notes<\/a> Chapter 5 Quadratic Equations to learn maths more effectively.<\/p>\n

TS 10th Class Maths Notes Chapter 5 Quadratic Equations<\/h2>\n

\u2192 The general form of a linear equation in one variable is ax + b = c.<\/p>\n

\u2192 Any equation of the form p(x) = 0 where p(x) is a polynomial of degree 2, is a quadratic equation.<\/p>\n

\u2192 If p(x) = 0 whose degree is 2 is written in descending order of their degrees, then we say that the quadratic equation is written in the standard form.<\/p>\n

\u2192 The standard form of a quadratic equation is ax2<\/sup> + bx + c = 0 where a \u2260 0. We can write it as y = ax2<\/sup> + bx + c.<\/p>\n

\u2192 There are various occasions in which we make use of Q.E. in our day-to-day life.
\nEg : The height of a rocket is defined by a Q.E.<\/p>\n

\u2192 Let ax2<\/sup> + bx + c = be a quadratic equation. A real number a is called a root of the Q.E. if a\u03b12<\/sup> + b\u03b1 + c = 0. And x = a is called a solution of the Q.E. (i.e.) the real value of x for which the Q.E ax2<\/sup> + bx + c = 0 is satisfied is called its solution.<\/p>\n

\u2192 Zeroes of the Q.E. ax2<\/sup> + bx + c = 0 and the roots of the Q.E. ax2<\/sup> + bx + c = 0 are the same.<\/p>\n

\"TS<\/p>\n

\u2192 To factorise a Q.E. ax2<\/sup> + bx + c = 0, we find p, q \u2208 R such that p + q = b and pq = ac.
\nThis process is called Factorising a Q.E. by splitting its middle term.
\nEg : 12x2<\/sup> + 13x + 3 = 0
\nhere a = 12; b = 13; c = 3
\na.c = 12 \u00d7 3 = 36
\nb = 4 + 9 where 4 \u00d7 9 = 36
\nNow 12x2<\/sup> + 9x + 4x +3 = 0
\n\u21d2 12x2<\/sup> + 9x + 4x + 3 = 0
\n\u21d2 3x(4x + 3) + 1 (4x + 3) = 0
\n\u21d2 (4x + 3) (3x + 1) = 0
\nHere 4x + 3 = 0 or 3x + 1 = 0
\n\u21d2 4x = -3 or 3x = -1
\n\u21d2 x = \\(\\frac{-3}{4}\\) or x = \\(\\frac{-1}{3}\\)<\/p>\n

\\(\\frac{-3}{4}\\) and \\(\\frac{-1}{3}\\) are called the roots of the Q.E. 12x2<\/sup> + 13x + 3 = 0 and x = \\(\\frac{-3}{4}\\) or \\(\\frac{-1}{3}\\) is the solution of the Q.E 12x2<\/sup> + 13x + 3 = 0.<\/p>\n

\u2192 In the above example, (4x + 3) and (3x + 1) are called the linear factors of the Q.E. 12x2<\/sup> + 13x + 3 = 0<\/p>\n

\u2192 We can factorise a Q.E. by adjusting its left side so that it becomes a perfect square.
\nEg : x2<\/sup> + 6x + 8 = 0 \u21d2 x2<\/sup> + 2. x. 3 + 8 = 0 \u21d2 x2<\/sup> + 2.x.3 = -8
\nThe L.H.S. is of the form a2 + 2ab
\n\u2234 By adding b2<\/sup> it becomes a perfect square
\n\u2234 x2<\/sup> + 2.x.3 + 32<\/sup> = -8 + 32<\/sup>
\n\u21d2 (x + 3)2<\/sup> = -8 + 9
\n\u21d2 (x + 3)2<\/sup> = 1
\n\u21d2 x + 3 = \u00b11
\nNow we take x + 3 = 1 or x + 3=-1
\n\u21d2 x = -2 or x = -4<\/p>\n

\u2192 Adjusting a Q.E. of the form ax2<\/sup> + bx + c = 0 so that it becomes a perfect square.
\nStep -1: ax2<\/sup> + bx + c = 0 \u21d2 ax2<\/sup> + bx = -c
\n\u21d2 x2<\/sup> + \\(\\frac{b}{a}\\)x = \\(\\frac{-c}{a}\\)<\/p>\n

Step – 2: x2<\/sup> + \\(\\frac{\\mathrm{bx}}{\\mathrm{a}}+\\left[\\frac{1}{2}\\left(\\frac{\\mathrm{b}}{\\mathrm{a}}\\right)\\right]^2=\\frac{-\\mathrm{c}}{\\mathrm{a}}+\\left[\\frac{1}{2}\\left(\\frac{\\mathrm{b}}{\\mathrm{a}}\\right)\\right]^2\\)<\/p>\n

Step – 3: \\(\\left(x+\\frac{b}{2 a}\\right)^2=\\frac{b^2-4 a c}{4 a^2}\\)<\/p>\n

Step – 4: Solve the above
\nEg: 5x2<\/sup> – 6x + 2 = 0 \u21d2 x2<\/sup> – \\(\\frac{6 x}{5}=\\frac{-2}{5}\\)
\n\"TS<\/p>\n

\u2192 Let ax2<\/sup> + bx + c = 0 (a \u2260 0) be a Q.E., then b2<\/sup> – 4ac is called the Discriminant of the Q.E.<\/p>\n

\u2192 If b2<\/sup> – 4ac > 0, then the roots of the Q.E. ax2<\/sup> + bx + c = 0 are given by
\nx = \\(\\frac{-b \\pm \\sqrt{b^2-4 a c}}{2 a}\\). This is called quadratic formula to find the roots<\/p>\n

\u2192 The nature of the roots of a Q.E. can be determined either by its discriminant or its graph.
\nQ.E.: y = ax2<\/sup> + bx + c.
\n\"TS<\/p>\n

\"TS<\/p>\n

Important Formulas:<\/p>\n