{"id":10496,"date":"2022-04-22T14:19:14","date_gmt":"2022-04-22T08:49:14","guid":{"rendered":"https:\/\/mathemerize.com\/?p=10496"},"modified":"2022-04-22T14:19:18","modified_gmt":"2022-04-22T08:49:18","slug":"find-the-zeroes-of-the-quadratic-polynomials-and-verify-a-relationship-between-zeroes-and-its-coefficients","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/find-the-zeroes-of-the-quadratic-polynomials-and-verify-a-relationship-between-zeroes-and-its-coefficients\/","title":{"rendered":"Find the zeroes of the quadratic polynomials and verify a relationship between zeroes and its coefficients."},"content":{"rendered":"

Question<\/strong> : Find the zeroes of the quadratic polynomials and verify a relationship between zeroes and its coefficients.<\/p>\n

(i)<\/strong>\u00a0 \\(x^2 – 2x – 8\\)<\/p>\n

(ii)<\/strong>\u00a0 \\(4s^2 – 4s + 1\\)<\/p>\n

(iii)<\/strong>\u00a0 \\(6x^2 – 3 – 7x\\)<\/p>\n

(iv)<\/strong>\u00a0 \\(4u^2 + 8u\\)<\/p>\n

(v)<\/strong>\u00a0 \\(t^2 – 15\\)<\/p>\n

(vi)<\/strong>\u00a0 \\(3x^2 – x – 4\\)<\/p>\n

Solution<\/strong> :<\/p>\n

(i)<\/strong>\u00a0 \\(x^2 – 2x – 8\\) = \\(x^2 – 4x + 2x – 8\\) = x(x – 4) + 2(x – 4)\u00a0 = (x – 4)(x + 2)<\/p>\n

So, the value of \\(x^2 – 2x – 8\\) is zero when x – 4 = 0 or x + 2 = 0. i.e. when x = 4 or x = -2.<\/p>\n

So, the zeroes of \\(x^2 – 2x – 8\\) are 4, -2<\/strong>.<\/p>\n

Sum of the zeroes = 4 – 2 = 2 = \\(-(-2)\\over 1\\) = \\(-coefficient of x\\over coefficient of x^2\\) = 2<\/strong><\/p>\n

Product of the zeroes = 4(-2) = -8 = \\(-8\\over 1\\) = \\(constant term\\over coefficient of x^2\\) = -8<\/strong><\/p>\n

(ii)<\/strong>\u00a0 \\(4s^2 – 4s + 1\\) = \\(4s^2 – 2s -2s + 1\\) = \\((2s – 1)^2\\)<\/p>\n

So, the value of \\(4s^2 – 4s + 1\\) is zero when 2s – 1 = 0 or s = \\(1\\over 2\\)<\/p>\n

So, the zeroes of \\(x^2 – 2x – 8x\\) are \\(1\\over 2\\), \\(1\\over 2\\)<\/b><\/p>\n

Sum of the zeroes = \\(1\\over 2\\) + \\(1\\over 2\\) = 1 = \\(-(-4)\\over 4\\) = \\(-coefficient of s\\over coefficient of s^2\\) = 1<\/strong><\/p>\n

Product of the zeroes = (\\(1\\over 2\\))(\\(1\\over 2\\)) = \\(1\\over 4\\) = \\(1\\over 4\\) = \\(constant term\\over coefficient of s^2\\) = \\(1\\over 4\\)<\/strong><\/p>\n

(iii)<\/strong> \\(6x^2 – 3 – 7x\\) = \\(6x^2 – 7x – 3\\) = \\(6x^2 – 9x – 2x – 3\\)<\/p>\n

= 3x(2x – 3) + 1(2x – 3) = (3x + 1)(2x – 3)<\/p>\n

So, the value of \\(6x^2 – 3 – 7x\\) is zero when the value of (3x + 1)(2x – 3) = 0 i.e.<\/p>\n

when 3x + 1 = 0 or 2x – 3 = 0\u00a0 i.e. when x = \\(-1\\over 3\\) or x = \\(3\\over 2\\).<\/p>\n

So, the zeroes of \\(6x^2 – 3 – 7x\\) are \\(-1\\over 3\\) and \\(3\\over 2\\)<\/strong>.<\/p>\n

Sum of the zeroes = \\(-1\\over 3\\) + \\(3\\over 2\\) = \\(7\\over 6\\) = \\(-(-7)\\over 6\\) = \\(-coefficient of x\\over coefficient of x^2\\) = \\(7\\over 6\\)<\/strong><\/p>\n

Product of the zeroes = (\\(-1\\over 3\\))(\\(3\\over 2\\)) = \\(-3\\over 6\\) = (\\(-1\\over 3\\))(\\(3\\over 2\\)) = \\(constant term\\over coefficient of x^2\\) = \\(-3\\over 6\\)<\/strong><\/p>\n

(iv)<\/strong> We have : \\(4u^2 + 8u\\) = 4u(u + 2)<\/p>\n

The value of \\(4u^2 + 8u\\) is zero when the value of 4u(u + 2) = 0 i.e. when u = 0 or u + 2 = 0 i.e. u = 0 or u = -2.<\/p>\n

So, the zeroes of \\(4u^2 + 8u\\) are 0, -2<\/strong>.<\/p>\n

Sum of the zeroes = 0 + (- 2) = -2 = \\(-(-8)\\over 4\\) = \\(-coefficient of u\\over coefficient of u^2\\) = –2<\/strong><\/p>\n

Product of the zeroes = (0)(-2) = 0 = \\(0\\over 4\\) = \\(constant term\\over coefficient of u^2\\) = 0<\/strong><\/p>\n

(v)<\/strong>\u00a0 We have, \\(t^2 – 15\\) = \\((t – \\sqrt{15})\\)\\((t + \\sqrt{15})\\)<\/p>\n

The value of \\(t^2 – 15\\) is zero when the value of \\((t – \\sqrt{15})\\)\\((t + \\sqrt{15})\\) is 0, i.e. when \\((t – \\sqrt{15})\\) = 0 or \\((t + \\sqrt{15})\\) = 0<\/p>\n

i.e. when t = \\(\\sqrt{15}\\), \\(-\\sqrt{15}\\).<\/p>\n

So, the zeroes of \\(t^2 – 15\\) are \u00a0\\(\\sqrt{15}\\), \\(-\\sqrt{15}\\)<\/strong>.<\/p>\n

Sum of the zeroes = (\\(\\sqrt{15}\\)) +\u00a0 (\\(-\\sqrt{15}\\)) = 0 = \\(-(-0)\\over 1\\) = \\(-coefficient of t\\over coefficient of t^2\\) = 0<\/strong><\/p>\n

Product of the zeroes = (\\(\\sqrt{15}\\))(\\(-\\sqrt{15}\\)) = -15 = \\(-15\\over 1\\) = \\(constant term\\over coefficient of t^2\\) = -15<\/strong><\/p>\n

(vi)<\/strong>\u00a0 We have : \\(3x^2 – x – 4\\) = \\(3x^2 – 3x – 4x – 4\\) = 3x(x +1) – 4(x + 1) = (x + 1)(3x – 4)<\/p>\n

The value of \\(3x^2 – x – 4\\) is zero when the value of (x + 1)(3x – 4) is 0, i.e. when x + 1 = 0 or 3x – 4 = 0,\u00a0i.e. when x = -1 or x = \\(4\\over 3\\)<\/p>\n

So, the zeroes of \\(3x^2 – x – 4\\) are -1, \\(4\\over 3\\).<\/strong><\/p>\n

Sum of the zeroes = -1 + \\(4\\over 3\\) = \\(-3 + 4\\over 3\\) = \\(1\\over 3\\) = \\(-(-1)\\over 3\\) =\u00a0 \\(-coefficient of x\\over coefficient of x^2\\) = \\(1\\over 3\\).<\/strong><\/p>\n

Product of the zeroes = (-1)(\\(4\\over 3\\)) = \\(constant term\\over coefficient of x^2\\) = \\(-4\\over 3\\)<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

Question : Find the zeroes of the quadratic polynomials and verify a relationship between zeroes and its coefficients. (i)\u00a0 \\(x^2 – 2x – 8\\) (ii)\u00a0 \\(4s^2 – 4s + 1\\) (iii)\u00a0 \\(6x^2 – 3 – 7x\\) (iv)\u00a0 \\(4u^2 + 8u\\) (v)\u00a0 \\(t^2 – 15\\) (vi)\u00a0 \\(3x^2 – x – 4\\) Solution : (i)\u00a0 \\(x^2 – …<\/p>\n

Find the zeroes of the quadratic polynomials and verify a relationship between zeroes and its coefficients.<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[43,910],"tags":[],"yoast_head":"\nFind the zeroes of the quadratic polynomials and verify a relationship between zeroes and its coefficients. - Mathemerize<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/find-the-zeroes-of-the-quadratic-polynomials-and-verify-a-relationship-between-zeroes-and-its-coefficients\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Find the zeroes of the quadratic polynomials and verify a relationship between zeroes and its coefficients. - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"Question : Find the zeroes of the quadratic polynomials and verify a relationship between zeroes and its coefficients. 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